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Full text of "Pressure dependence of magnetic transitions"

PRESSURE DEPENDENCE OF 
MAGNETIC TRANSITIONS 



By 
JAMES EDMUND MILTON 



A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF 

THE UNIVERSITY OF FLORIDA 

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE 

DEGREE OF DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 
April, 1966 



ACKNOWLEDGMENTS 

.-^ The author is deeply indebted to many people for 

-^ aid and encouragement in completing this dissertation. He 

wishes to express his gratitude to the many people who have 
helped and especially to those listed below. 

Dr. Thomas A. Scott, the chairman of the author's 
supervisory committee, suggested the problem and was always 
ready to give aid when needed. 

I'ir. Basil McDowell was absolutely indespensible, 
he assisted in the design and construction of much of the 
^,i equipment used in this study. In addition to this he sup- 
plied liquid helium when needed and frequently gave a 
friendly helping hand or clever suggestions. 

Dr. William S. Goree gave many helpful suggestions 
regarding the cryogenic apparatus and designed the pressure 
bomb used. 

Mr. K. S. Krishnan served as a soundingboard for 
many ideas involving experimental techniques and frequently 
7^) suggested improvements. 

I»lr. Guy Ritch and Mr. Frank Ebright were very able 
assistants during the final phase of data taking. 

Dr. Stanley S, Ballard and Dr. Thomas A. Scott 
provided financial support during most of the author's 



11 



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2) 



graduate career. Dr. Knox Millsaps graciously lightened 
the author's work load during the writing of this disser- 
tation. 

Deadlines would not have been met had it not 
been for the encouragement and invaluable help given by 
Dr. Richard L. Fearn in the preparation of this disser- 
tation. 

Finally the author wishes to thank I<Irs. Jacqueline 
Ward who very cheerfully and untiringly typed this manu- 
script. Mr. Guy Hardee lent an able hand in drawing the 
figures. 

This work was financed by Grant No. NSF-GP1866 
from the National Science Foundation. 



ill 



; 



) 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGMENTS H 

LIST OF FIGURES V 

LIST OF TABLES vil 

CHAPTER 

I INTRODUCTION 1 

II THEORY 22 

III APPARATUS AND PROCEDURE" $Li- 

IV RESULTS AND CONCLUSIONS 9^4- 

REFERENCES 120 

BIOGRAPHICAL SKETCH 125 



iv 



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LIST OP FIGURES 

Figure Page 

1. Rare earth crystal structures, 10 

\ 2. Magnetic structures of rare earths, 13 

3. Spontaneous magnetization, Weiss theory, 28 

^. Hydrogen molecule. • 30 

5. V/iring diagram for the inductance "bridge, 58 

6. Inductance "bridge, So 

7. Equivalent circuit for the inductance bridge, 63 

8. Entrance to high-pressure room. 68 

9. Top view of high-pressure room, 69 

10, Schematic of high-pressure system, 72 

11, Inside view of high-pressure room, 7^ 

12. Schematic of the control panel for the 

pressure system. yS 

13. Control panel, 78 
1^, High-pressure gas bomb, 81 
15' Electrical seal, 8^ 

16. Bomb plug and seals, 86 

17. Gryostat, 88 
^. 18. Inductance versus temperature for dysprosiijim, 96 

19. Neel transition for dysprosium. 97 

20. Pressure shift of Neel transition for dyspro- 
sium, 98 

21. Pressure shift of N4el transition for erbium, 100 



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22. Robinson «s interaction curve for rare earths. 101 

23. Interaction curve for rare earths. 103 
2^. - '^: '^ '/ for rare earths, 10^^ 
25. , Dysprosium data. I07 

26. Pressure shift of Curie transition for 
dysprosium. IO8 

27. Erbium data taken while cooling. 109 

28. ErbiuEi data taken while warming, no 

29. Pressure shift for middle peak on warming 

data for erbium. 112 

30. Pressure shift for upper peak on warming data 

for erbium. , 11 3 

31. Pressure shift for upper peak on the cooling 

data for erbium. 114 

32. - ^I'^Tf f<^3r heavy rare earths. II5 
a m V 



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■-- -*-— ^— «• 



) 



LIST OP TABLES 
Table Page 

1 Sunmary of available experimental information 

on rare earth magnetic structures. 14 

2 Summary of experimental results on the pres- 
sure shifts of the magnetic transitions for 

the rare earths* 20 

3 Sample coils. 64 



vii 



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CHAPTER I 

INTRODUCTION 

Historical baolcg:round 

Magnetism has aroused man's curiosity and fired 
his imagination for at least several thousand years. 
References to the attractive power of the lodestone had 
already appeared in Greek X'jriting 'bj 600 B.C.-^ The first 
knoim reference to the fact that magnetism could also 
repel a body is found in the writings of the Roman 
Lucretius Carus in the 1st Century B.C. 2 it is interesting 
J that no references to the directive property of the magnet, 

as used in the compass, is found in old Greek and Roman 
literature, but beginning in the period 1000-1200 A.D. the 
history of magnetism is closely associated with the compass 
and its use in navigation. The first clear mention of a 
magnet used to indicate direction xms made by Shen-Kua 
(IO3O-IO93), a Chinese mathematician and instrument maker. 
By 1100 A.D., the Chinese Chu lu reports that the compass 
was in use by sailors going betvjeen Canton and Siimatra. In 
1269, Peregrinus de Maricourt reported on experiments made 
on a spherical lodestone. 3 He explored the surface of the 
sphere x^rith small particles of iron and applied the term 
pole to the places in x\rhiGh the magnetic power appeared to 



■ .■n"'iiii iufi •i.i"*« 



be concentrated. Little progress is reported until I6OO 
when V/llliam Gilbert published his De Ma?;:nete which sum- 
marized the knowledge of magnetism and reported the 
results of many of his 0^;^.! experiments. In this book he 
propounds his own great contribution, the realization that 

^ the earth itself is a magnet. Also of great importance is 

his determination that if magnetized iron is heated to a 
bright red it loses its magnetism. 

In 1785, Charles A. Coulomb^ established with 
some precision the inverse square law of attraction or 
repulsion between unlike and like magnetic poles. This 
result was taken over by Poisson who became the best inter- 
preter of the physical constructs which Coulomb had dis- 

ij covered. -5 To magnetism, Poisson brought the concept of the 

static potential, which had been so successful in solving 
the problems of static electricity. He also assumed mag- 
netization to be a molecular phenomenon, but believed that 
the molecule became magnetic only when the two fluids It 
contained became separated. It was Weber° who proposed 
that each molecule is a permanent magnet, subject to a 
frictlonal force that tends to maintain it in its established 

orientation. This theory failed to explain the existanoe 

) 

01 residual induction and hystersis. 

In 1820, Oersted discovered that an electric 
current would affect a magnetic needle. Ampere then inves- 
tigated experimentally and mathematically the forces 
between currents. He was able to show that a current in a 



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3 

circuit was equivalent to a magnetic shell of calculable 
strength. He then put forth the hypothesis that magnetism 
arose from currents within the molecules. This theory 
stood until it was modified by modern quantum mechanics. 
Pierre Curie made the first extensive study of 
) . ^he thermal properties of magnetic materials. 7 As a 

result of these studies he was able to establish that the 
magnetic susceptibility of a paramagnetic material was 
inversely proportional to the absolute temperature. 

X = C/T (1.1) 

The constant of proportionality was determined for each 
material and was always found to be positive. He found a 
^relatively rapid decrease in the magnetization as each 
ferromagnetic material was heated to a critical tempera- 
ture, now called the Curie temperature. Above this 
temperature, which was different for each material, the 
behavior was much like an ordinary paramagnetic substance. 

The first important modern development in mag- 
netic theory came when Langevin^ used statistical mechanics 
to derive the Curie law, equation (1.1). The underlying 

assumptions of his theory were that, each molecule had a 
) 

definite magnetic moment that tended to be aligned by the 

applied magnetic field and at the same time disturbed by 

thermal agitation. I-Iany years later this derivation was 

modified by Brillouin who took into account the quantum 

. mechanical requirement that the atomic magnetic moments Were 



mAa-^'HliMir^t^i 



4 

restricted to a finite set of orientations relative to the 

applied field. 9 

After the work by Langevin, Weiss^^ made the 
next big step in developing a modern theory of magnetism. 
He assumed that the molecules are exposed both to the 

\ applied field and to a so-called molecular field propor- 

tional to the magnetization. It is a consequence of the 
Weiss theory that small regions, domains, within a mag- 
netic material are magnetized to saturation even though 
the net magnetization of the body is zero. This is 
possible by having the magnetic moments of the domains 
oriented randomly. The Weiss theory is a very successful 
one ^^^hich has been substantiated by many experiments. It 

v~N ls» however, not a very pleasing one since no explanation ' 
of the origin of the molecular field is given. 

In 1925 » Uhlenbeck and Goudsmit introduced the 
concept of electron spin to explain some discrepancies 
between theory and experimental measurements of the spectra 
of one-electron atoms and the alkali -metals. This spin has 
associated with it both an angular momentum and a magnetic 
moment, Follox^jing this came the enunciation of the Paul! 
esiclusion principle. These two developments allowed Dirac 
and Heisenberg^l»^2 ^q demonstrate a quantum mechanical 
origin for the Weiss molecular field. In their theory one 
starts with the Heitler-London model of the hydrogen mole- 
cule and considers a Hamiltonian made up wholly of electro- 
static terns and kinetic energy terms. The Pauli exclusion 



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principle enters the discussion only through symmetry 
requirements. Further, one must make an assumption 
regarding a distribution of energy levels. Heisenberg 
assumed this distribution was Gaussian. Using these 
conditions, it was possible to reproduce the Weiss theory 
^ and show that the origin of the molecular field was a 

quantum mechanical ezchange integral. Thus, the most 
successful theory propounded had been given a quantum- 
mechanical basis and indeed it seemed that magnetism might 
at last be understood. But, as is usually the case, 
things were not as good as they appeared. Heisenberg's 
theory suffered from several serious weaknesses. 1) It 
was based upon the hydrogen molecule and contained no 
account of lattice periodicity. 2) The results obtained 
were very much dependent upon the distribution of energy 
levels assumed. 3) The actual calculation of the exchange 
parameter was an extremely difficult problem and thus far 
has not been resolved. 

At about the same time the Heisenberg-Dirac 
theory was being developed, Ising^3 proposed a different 
method for looking at the problem. The spins were disposed 
at regular intervals along the length of a one-dimensional 
chain. In accordance with the laws of Uhlenbeck and 
Goudsmit each spin was allowed to take on only one of two 
possible orientations. It was possible to obtain an exact 
solution for this model if it was assumed that each spin 
Interacted XNrith only a finite number of neighbors. 



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Unfortunately the result indicated that ferromagnetism 
should not occur above 0°K. Since that time exact two- 
dimensional solutions and approximate three-dimensional 
solutions have given finite transition temperatures, thus 
showing that the failure of the first model was due to 
\ its dimensionality. 

Various methods have been employed to try to 
improve on the Heisenberg theory. One of the most suc- 
cessful of these is the method of spin waves developed by 
Bloch and Slater. ^^'^^ This theory starts with the 
observation that the eigenvalue of the Heisenberg exchange 
coupling can be determined rigorously if the spins of all 
but one atom are parallel. Furthermore, approximate 
solutions can be found if the number of reversed spins is 
small when compared with the number of atoms. Due to the 
above assumption this theory is good only for very low 
temperatures. It has been quite successful in describing 
the variation of magnetization with temperature in this 
region. 

In 1932, Neel proposed a theory to account for a 
type of paramagnetic susceptibility temperature dependence 
which did not agree with any of the existing theories.!^ 
He proposed two interpenetrating sublattices undergoing 
negative exchange interaction. This theory continues to be 
the basis for modern developments in the theory of what is 
noi-r called antiferromagnetism. 



i 



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I 7 

The rare earths, which have become available in 

quantity in pure form only since the development of the 
ion exchange method of separation, cannot be described 

, completely by the theories discussed in the preceding 

section. They have spurred a renewal in interest in 

J magnetism on both the experimental and theoretical fronts. 

Their physical properties v;ill be discussed in detail in 
the following section and the theories developed to 
describe them will be examined in a later chapter. 

Structure and information on rare earths 

The rare earth metals are composed of the fifteen 
elements which range from lanthanum to lutetium. The elec- 
j- .. tronic structure of these elements is normally given by 

(4f)^(5s)2(5p)6(5i)l(5s)2 

where n increases from to 14' as the atomic number in- 
creases from 57 to 71. The outer electronic structure, 
which essentially determines their chemical properties, is 
the same for all of these elements and they normally appear 
in compounds as tripositive ions. Often scandium and 
. yttrium, atomic numbers 21 and 39 respectively, are grouped 
with the rare earths since their external electronic con- 
figuration is similar. ■'■'^'■'■° 

The k-f electrons are tightly bound inside the 
outer closed shells on the atoms and therefore play only a 
small role in chemical bonding. They behave almost as they 



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8 

would in a free ion, giving a resultant angular moraentum 
due to both spin and orbital motion. Since there is a 
magnetic moment associated with this angular momentum, all 
rare earth compounds have interesting magnetic properties. 
It will be shown in Chapter II that a model 

J ' based on ions with ^f^ configurations acted on by crystal- 
line fields and coupled by exchange interactions is 
capable of explaining much of the magnetic phenomena of 
the rare earth metals. 

Rare earths have been investigated extensively 
within the last ten years at the Ames Research Laboratory 
and the Oak Ridge National Laboratory. The former of 
these has been involved in the separation and purification 

j- of these elements, and with the measurement of specific 

heat, thermal expansion, electrical resistivity, magnetic 
properties, and other physical properties. The latter 
group has performed neutron diffraction studies and 
determined the complicated magnetic structures. 

The tripositive ion picture, outlined above, is 
violated by two of the rare earths, Eu and Yb, which come 
immediately before the middle element and the last element 
of the series, respectively. These elements should have 
■'^f and 4f -^ configurations, however, they appear to 
prefer to gain the extra correlation energy of a half- 
filled or completed shell and take a divalent form with 
^f"^ and ^^f configuration. Ytterbium has only a small 
paramagnetism as would be expected from a closed shell and 



9 

europium shows large magnetic moments as it should for 

the kt' structure. One other element that should be men- 
tioned in this respect is Ce, which may be found in a 
four-valent state either at low temperature or at high 
pressure. This is due to the fact that at the beginning 
J of the series the ^f and 5d. electrons have similar energies. 

In this state, as would be expected, Ce is found to 
exhibit small paramagnetism. Above lOO^K the stable form 
is found to be ^f^ and trivalent. 

The room temperature crystal structure of the 
rare earths tend to fall into two categories, the hexa- 
gonal close packed and a double hexagonal structure as 
shown in Figure 1. While they have been reported with 
various crystal structures, it seems that La, Ce, Pr and 
Nd, the light rare earths, usually have the modified 
hexagonal structure. Promethium has no stable isotope 
and therefore no information is available. Next, Sm has 
a very complicated hexagonal structure which repeats after 
nine hexagonal layers. The remainder, Gd, Tb, Dy, Ho, Er, 
Tm and Lu, have hexagonal close packed structures with c/a 
ratios 1.57-1 •59- Their magnetic properties, while complex, 
show a certain regularity which may be traced to exchange 
interactions and crystalline fields. 

In the absence of detailed knowledge of the band 
structure of any of these elements theoretical work has 
been based upon the crude approximation of nearly free 
electrons. The effect of lattice symmetry has to some 



.; 



10 




B 




B 



Hexagonal close packed 
structure 



Double hexagonal structure 



Figure 1, Rare earth crystal structures. 



11 

extent been included by considering the Brillouin zone 
structure!? for the heavy rare earth series. The primi- 
tive translations t, , U and ^3 are shown in Figure 1. 
There are two atoms per unit cell, one at the origin and 
one at T= -^(t,+ 2t0-f ^ ^^ . The reciprocal lattice is 
also hexagonal and has vectors X. and T» with magnitude 
a/T ' ^^ the basal plane 120° apart, and ^3 perpen- 
dicular to this plane with magnitude 2tt/c . 

EvQn when the atoms are triply ionized the 
tightly bound ^f electrons are shielded from the crystal- 
line field by the Ss^ and 5p° electrons. Under these 
conditions their orbital angular momentum remains 
unquenched by the fields of neighboring ions. These 
electrons have total orbital angular momentum, L, and 
total spin, S, in the ground state as prescribed by 
Russel-Saunders coupling and Hund's rules. The energy 
difference between the ground state J multiplet and the 
first excited J multiplet is usually greater than 0.1 ev, 
therefore the excited multiplet plays no role in thermal 
properties. 

There are basically four types of measurements 
that have been made, in order to determine the magnetic 
properties of these elements. They are neutron diffraction, 
bulk magnetic measurements, specific heat measurements, 
and electrical resistivity measurements. A brief dis- 
cussion of the information that can be obtained by each of 
these methods is given below. 



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12 

Detailed neutron diffraction studies have been 
carried out on several of the rare earths. 2'^»2-^» 22,23. 2^,25 » 26 
The magnetic structures thus determined have been found to 
be quite complex. These studies have also given infor- 
mation about the magnitude of the ordered moment and its 
temperature dependence. The magnetic properties are found 
to be highly anisotropic, that is, the moments along the 
c-axis are quite different from the moments in the basal 
plane. Figure 2 shows some of the types of ordering that 
have been found and Table 1 gives the transition tempera- 
tures and structures for each element. 

Bulk magnetic measurements have shown that the 
susceptibility of these elements at high temperatures is 
roughly described by the Curie-V/eiss law. 

^ 3K(T-<9) 

Here the Weiss constant, & , indicates the approximate 
value of the exchange energy. It is also possible to 
obtain the magnitude of the ordered moments from this type 
of experiment. If a sufficiently strong magnetic field is 
applied to one of the anti ferromagnetic structures it is 
possible, in some cases ; to change it to a ferromagnetic 
structure. The field at which this occurs is called the 
critical field and can be used to obtain information about 
the energy difference between the tvro states. 



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15 

The transitions from one magnetic state to 
another are accompanied by sharp peaks in the specific 
heat versus temperature curves. These peaks can be located 
very accurately and therefore allow accurate determination 
of transition temperatures. 

The electrical resistance of these elements show 
anomalies at the magnetic transitions. These anomalies 
have been used to locate the transitions by a number of 
investigators. 

Experimental XTork 

The way in which magnetic properties of materials 
vary with pressure has long been of interest to physicists. 
Some of the earliest experiments in this area were done in 
an attempt to gain information about the origin of the 
earth's magnetic field. ^'^ The more recent ones, however, 
have been done in order to try to obtain information on 
the volume dependence of the exchange integral. One of the 
methods of attack on this problem has been to measure the 
shift X'l'ith pressure of the temperature at which the material 
goes from one magnetic state to another. The temperature 
of the transition to the ferromagnetic state is called the 
Curie temperature, 7c, while the temperature of transition 
from the paramagnetic to the anti ferromagnetic state is 
called the Neel temperature, Tn • The discussion which 
follows will be confined to experiments in which d T/d P 
have been determined. There is a vast literature of other 



0. 



X 



16 

types of magnetic experiments including a recent review 

by Ko-uvel.28 

One of the earliest attempts in this field was 
that of Yeh in 1925.29 Yi% measured the effect of pres- 
sure on the magnetic permeability of iron, nickel and 
cobalt. This work was followed by that of Steinberger 
who, in 1933 ». d-id essentially the same experiment with 
improved sample annealing techniques ^^ These experiments 
did not specifically set out to measure the shift of the 
magnetic transition temperature with pressure. In retro- 
spect, however, it can -be recognized that Steinberger 
actually induced a phase change from the ferromagnetic 
state to the paramagnetic state in a 30Ni 70Fe sample by 
the application of pressure at room temperature. From 
his data it can be concluded that d 7c /d P < O for this 
alloy. 

The first actual attempt to measure dlc/dP 
was made in I931 by Adams and Green^? who studied iron, 
nickel, magnetite, nickel steel and meteoric iron. They 
used the transformer method for detecting the transition. 
A primary and a secondary coil were wound on a closed 
frame made of the sample material. An alternating voltage 
was applied to the primary and the' output voltage was 
monitored as a function of temperature. The drop in out- 
put at Tc is very sharp, and although it does not define 
the Curie temperature in the conventional way, the method 
is satisfactory for finding a change in Curie point. They 



V 



17 
used carbon dioxide as the pressure transmitting medium 
and therefore achieved truly hydrostatic pressure. The 
shift for pressures up to 3.5 kilobars was found to equal 
zero for all of their samples. This result has not been 
confirmed by other investigators and it is believed that 
thermal uncertainties masked the true changes. 

Michels et al.^l used the discontinuity In the 
0/RWdR/dT) versus temperature curve to indicate Tc • 
The sample material, which was 70Ni 30Cu, exhibited a 
broad transition that occurred gradually over 50°C. 
They concluded from this that it was necessary to deter- 
mine a shift of the Curie region. By carefully analyzing 
their data they were able to obtain dTc /dP — +6.4x/o"' 
°K/kilobar. This method requires very accurate resistance 
measurements over large temperature intervals and is com- 
plicated by the fact that resistance also changes with 
pressure. Later a monel alloy was studied by the same 
method. ^^ This transition also was quite broad and 
yielded d Tc/d P = 3x/0'^ °K/kilobar. 

Ebert and Kussman-^-^ used large magnetic fields to, 
obtain magnetization versus temperature curves so that the. 
Tc could be determined in the conventional manner. 
They then applied pressure and tried to determine dTc/dP 
for several pure metals and alloys. The result obtained for 
all samples was dTc/dP- O . Michels and De Groot^^ 
criticized their result and showed by a thermodynamic 
treatment of second order phase transitions that in general 



J 



18 

<^TcMP^ O , They further showed that the experi- 
mental method used by Ebert and Kussman was not accurate 
enough to show small but significant variations of Tc . 
Kornetzki35 took the data obtained by Ebert and Kussman 
and re-analyzed them and obtained non-zero values for 

d Tc /d P . 

In 195^. Patrick36 j^ade a detailed study to 
determine dH/dP of nickel, gadolinium, cobalt, iron, 
eight metallic alloys, a ferrite and a perovskite. The 
transitions were detected by the transformer method as 
developed by Adams and Green. Two pressure systems were 
used, one used a gas for the pressure transmitting medium 
and the other used a liquid. The pressure was truly 
hydrostatic. Patrick's results agreed with those of 
Michels et al. , and are widely quoted in the literature. 

Samara and Glardini37 made measurements on the 
shift of Tc in nickel and a nickel iron alloy. A 
multianvil pressure system with pyrophyllite as the 
transmitting medium was used. Pressures up to 35 kilobars 
were generated and the shifts found were in general agree- 
ment with those already determined. The transition was 
detected by monitoring the self -inductance of a coil which 
was wound on the sample. 

In addition to the electrical resistivity, self- 
inductance and transformer methods of detecting magnetic 
transitions, there are two other techniques which have been 
used. These transitions can be located by monitoring the 



19 

mutual inductance betx\reen two coils wound on the sample. 

Changes in the magnetic moment of the sample show as a 
change in this mutual inductance, which can be measured 
very accurately by bridge methods. Finally, when the 
pressure system permits, a method involving the extraction 
of the sample from a magnetic field can be used. 

The pressure systems for this type of study fall 
into two distinct categories, those whose pressure trans- 
mitting medium is a liquid or a gas and those that use a 
solid for pressure transmission. The former, of course, 
are the only ones which produce truly hydrostatic pres- 
sures, however, the latter are able to obtain much higher 
pressures. One study has been made by sealing water in 
the pressure vessel and then freezing the entrapped water. 
The disadvantage of this method is the possibility of 
having tremendous pressure gradients inside the pressure 
vessel. 

The results of all of the investigations of 
pressure shifts of the transition temperatures for pure 
rare earths as well as the pertinent information about 
methods used are summarized in Table 2. 

This dissertation deals with the effect of pres- 
sure on the Curie transition and the Neel transition in 
dyprosium and erbium. These experiments are the second in 
a planned series of high pressure studies to be carried 
out at the University of Florida. It was necessary to 
develop the complete pressure system as well as the methods 



20 



o 

0) 



VO 00 OnO 



co o^o 



H CM CO 0^ O C*-\ C^j:}- 



CO o 



J 



o 

M 
\ 

o 



CM 

00 



^ o o o\o 

U^\OVO^VO 

I I I I I 



^ 
^ 



o 

H 
•H 

O 



COMD O CM 


r-i ^\0 O- 


• • • • 


H rH tH H 


till 



O O 

• • 

I I 



^CO o 


00 


• • • 


-^ 


r-i 1 C^ 


• 


+ 1 


1 



CM 
i-i 



o 

+^ o 
o ^ 

<D -P 

+i © 
<u 6 



M-p ^ ^ 


•P fH in 


<D o e 


o <D <y 


S p:S S S 


us Ei 


fH Id ?H fn 


-C) ?H fH 


O s O o 


COO 


<tH M Cn (i-H 


M Cm C(H 


w ra Ki 


W CO 


SH ' fi ^ 


• fl C 


cd -P OS 05 


+:> c6 OS 


fH JS iH h 


7i ^ h 


EH S EH EM 


S Eh EH 



-P ^ 'm ?-i i-\ U +ifs 

;so;3assao ^a 

SIJcJCOOOO-P CO 

H-PH<(-(<tHCM<iHO M<(-i 

CO CO CO CO CO Cd CO 

. ^ . fl C fl C iH . C 

-Pto4^aia5c6oJ-P +:>o3 

?S0:3hf-iM!HW ^^) 

aWSBEHEHEHW g:EH 



<0 



in S 

CO tH 
CO Ti 

fH a 







^^ '1 




to to H H 


CO H H 


pi-H tOHHHH O 


CO H 


cs5 cS O O 


03 O O 


tA r-\ CflOOOOPQ 


OS U 


C5 CiJ M bO 


O bO 60 


-d O O bO bO &0 bO 


Ci bO 


<! < 


<< 


C CO <Ji <t! <! <! <D 

M o 

HI 


^^ 



a) 

(H 






CO 

to bOrO 

CO c o 
0) 03 H 



■ri- CO 

C^CM ^^(H -^ VPvC^ >^ . CM 

CO\D r^V^ ^ Cvl CN- CO -* ^ CM £V O- CM rH MD CO 

! i I I III I 1 t I I I I I II 

o o o lA ooo o u^o o xpvo iN-o ovr\ 



-p 
© 
© 





;3 



H 
O 

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© 

Eh 



a 

■H 
«1 
O 

^H 
ft 
CO 

!>> 
Q 



a 

»H 

a 

H 
O 



21 
and equipment necessary for performing the experiments. 
A large high-pressure helium gas system, which is des- 
cribed in detail in Chapter III, was constructed. A gas 
system was chosen in order to be able to work under truly 
hydrostatic pressure at low temperatures. The choice of 
samples was based upon the availability of high purity 
specimens and also the desire to take advantage of the 
ability of the pressure system to work at very low temp- 
eratures. l^^len this work was started there were no 
published results on pressure shifts in any of the rare 
earths. As can be seen in the preceding section there has 
recently been a flurry of activity in this field. The 
pressure system constructed here is still the only one 
capable of studying the lower temperature transitions 
imder hydrostatic conditions and further studies on 
holmium and thulium are underway presently. 

The results obtained for erbium for both 
6Tn/dP and dTc/dP are new. The results obtained 

for d Tn /d P for dysprosium are presented as corrob- 
orating those which have now been published. The 
d Tc /d P for dysprosium is in marked disagreement with 
that presented by Robinson et al.^^ which is the only one 
published to date. A complete discussion of the results 
is given in Chapter IV, 



CHAPTER II 
THEORY 



^ 



Introduction 



Any discussion on magnetism must be based on 
quantum mechanical concepts. In the general discussion 
of magnetism which follows the author has relied heavily 
on numerous references. ^5-53 The discussion of rare 
earths mainly follows the reviews by Elliott, ^^ Yosida-5-5 
and the books by Van Vleck-^o and Chikazumi.-5'I' 

This discussion can in no way be thought of as 
complete, but rather will attempt to describe the methods 
that have been most successful in treating the problem 
of magnetism. While much progress has been made there 
exists, at present, no completely satisfactory theory. 

Types of map:netism 

This section begins with a brief summary of the 
types of magnetism that are observed and some remarks 
concerning their origin- The major classifications are 
diamagnetism, paramagnetism, ferromagnetism, antiferro- 
magnetism and ferrimagnetism. 

Diamagnetism is a weak magnetism in which a 
magnetization is exhibited opposite to the direction of 



22 



23 

the applied field. It is associated with the tendency 
of electric charges to" shield the interior of a body from 
an applied magnetic field. It can be looked upon as a 
manifestation of the well-known Lenz's law, which states 
that when the flux through an electrical circuit is changed 
an induced current is set up in such a direction as to 
oppose this flux change. In a resistanceless circuit 
such as the orbit in an atom or in a superconductor the 
induced current persists as long as the field is present. 
Landau58 has shown that there can also be a diamagnetic 
contribution from the conduction electrons in a metal. 
Diamagnetism is present in all substances; however, in 
all cases except the superconductor it is a small effect 
with a susceptibility on the order of -10-5 cm3/mole. 
This effect is swamped if any other type of electron 
magnetism is present. The superconductors, which exclude 
all magnetic fields, exhibit perfect diamagnetism and have 
a susceptibility equal to -l/kn . Diamagnetism plays a 
small role in the rare earths and will not be mentioned 
in the remainder of this discussion. 

Paramagnetism arises in materials in which there 
are permanent magnetic moments present. Magnetization 
results from the orientation of these moments in an 
applied field. This orientation is opposed by thermal 
agitation and therefore would be expected to be highly 
temperature dependent. The permanent moments may arise 
from the spin and orbital motion of the electrons or from 



jWifli*^jMp<i, ' > I 



21^ 
the nuclei. In the rare earths the nuclear susceptibility 
is about 10-^ times the electron susceptibility and will 
not be considered in this discussion. The electron para- 
magnetic susceptibilities vary from about +10-5 to 
+10-2 cm3/mole. The rare earths are accurately described 
) in the paramagnetic region by the Curie-Weiss law which 

will be developed in a following section. 

A substance is called ferromagnetic if it pos- 
sesses a spontaneous magnetic moment even in the absence 
of an applied magnetic field. This moment occurs only 
below some critical temperature known as the Curie tempera- 
ture. This type of behavior is explained by adding to the 
paramagnetic model a strong co-operative effect which tends 
j - ■to align the permanent moments in a parallel manner. 

Si^°e dysprosium and erbium are both ferromagnetic at low 
temperatures the theories of this type of magnetism will 
be discussed in a following section. 

Antiferromagnetism arises from co-operative 
effects in a manner similar to ferromagnetism. In this 
case, however, the spins tend to align themselves in an 
antiparallel manner. The net magnetization is small and 
gives susceptibilities on the order of the ones given by 
paramagnetism. The temperature dependence of this suscep- 
tibility is, however, very different from that of para- 
magnetism. More will be said about this phenomenon in 
following discussions. 



Mae>4CMil«i^Me 



25 

The oldest magnetic material known, the lodestone, 
is a ferrimagnet. This type of magnetism is characterized 
by an antiparallel arrangement of moments "but with the 
moments of unequal magnitude. This can give a strong 
external magnetic field. This type of effect is thought 
to arise from the same type of interaction as the anti- 
ferromagnetic case. These materials are of great prac- 
tical Importance since many of them are insulators. None 
of the rare earths exhibit this type of magnetism so no 
further mention of it will be made. 

Quantum mechanical Langevin theory of paramagnetism 

Consider a system of M independent atoms in a 
magnetic field H • There will be 2 J + I Zeeman levels 
for each J . Assume that, as with the rare earths at room 
temperature, kT is small compared to the energy gap 
between the ground state and the first excited state J"' . 

V/rite the operator equation "p - ^ Pt>J where 
jUfj = efi/(2mc)is the Bohr magneton and a is the Lande q 
factor given by 

The energy of interaction between the magnetic moments and 
the applied field is given by W(h) = -p- H = "O l;(j MtH 
Using statistical mechanics it can be shown that the 
magnetic susceptibility is given by X= 4t^ ^^'" ^^ , 
Where 2 =fS~"' is the partition function for the system. 



26 



In this system 



2 = 



"lexp ^^i^f 



Mt = -J 



which gives 



x = 



NkT 
H 






H' 



After some mathematical manipulation one obtains 

for the magnetic susceptibility, where Bj-(x) is 
called the Brillouin function and can be written 

B.,,= ^ coihiWU -27 cothiif) 



(2.1) 



where 






(2.2) 



If the energy of Zeeman splitting is small compared to 
kT then X«l and one obtains a nearly equal 
probability of occupation for all levels. Under these 
conditions Bjo^) can be expanded In a power series 
and higher order terms neglected to obtain 



B,7 



{y) 



3J 



X 



(2.3) 



By using equations 2.1, 2.2 and 2.3 one cdn obtain 



27 



X= Nr(J^^Oq>' (3.,, 



It can be seen that equation 2,k is equivalent to 
equation 1.1» the Curie law, where C = N J( J+l)g'^p/3 k . 

By using equation 2.1 and the relationship 
|V1=XH a general expression for the magnetization of 
a material obeying the above theory can be written as 

In the special case where X« 1 equation 2.1 and 2A 

can be combined to obtain 



J K I 



I Weiss theory of ferromap:netism 

Weiss modified the above theory by adding to 
the paramagnetic model an interaction which tended to make 

I the atomic moments align themselves in a parallel manner. 

^ He defined a molecular field proportional to the magnl- 

tization of the sample, Hm = ^M. where V is the Weiss 
constant. Now, using the methods of the previous deri- 

I 

vation, one can obtain some useful relationships. If the 
magnetic field in equation 2.2 is replaced by an effective 



J 



y 



28 
field, Heff =■ H + 2f M , then for ferromagnetic materials 

X= ^^fH^-^M) (2.7) 

In order to look for the spontaneous magnetization let 
1-1 =• O and solve equation 2.? for M to obtain 



Since M must satisfy both equations 2.5 and 2.8 the 

simplest procedure is to investigate its behavior at 

various temperatures by graphical methods. 
M 



(2.8) 




Figure 3« Spontaneous magnetization, Weiss theory. 

From Figure 3 it can be seen that there is a critical 
temperature, Tc , below which one gets spontaneous 
magnetization due to the -^ M field. As the temperature 
increases through Tc this magnetization vanishes. 
From equations 2.5 and 2.8 it is possible to obtain the 
follomng expression 



3 



__ XkT 



where it has been assumed that X« I 



By talcing the 



29 

derivative with respect to X of both sides of the above 
equation and evaluating it at T= Tc and X — O it is 
possible to obtain the following relation between the 
Curie temperature and the Weiss constant. 



-r. _ N9Vj(J-H)?r (2.9) 



Let us now consider a temperature region above 
Tc so that there is no spontaneous magnetization. 
Then equation 2.5 becomes 

M= Ng>,%m^)) (H4-j^M) (2.10) 

By using equations 2.9 and 2.10 it is possible to obtain 

(2.11) 



T-Tc 

where C = Tc// • Equation 2.11 is known as the Curie- 
Weiss law. 

This is a very successful phenomenological 
theory which describes accurately the results of many 
experiments. In deriving it Weiss made no attempt to 
explain the origin of the molecular field. 

Heisenberg-Dirac theory of ferromagnetism 

Heisenberg was the first to show that the Weiss 
local field could be given a quantum mechanical origin. 
This can be demonstrated by considering the Heitler-London 



30 

solution for a hydrogen molecule. Consider a simple 
system of two atoms, a and b , that have one electron each 
and are separated by a distance /ab. See Figure 4. 




Figure 4-, Hydrogen molecule. 
Consider the following Hamiltonian, 

H= T, --f- +Ta— ^ ^■¥- ^-T- -¥^--T^ (2.12) 

roi fba fab Til laa Tbi 

where "T denotes the kinetic energy operator and the 
subscripts identify the electrons. Consider also the 
following relationships. 

(t, --^,)la(o> = £laio> 

(T--^)lbc,,>=elU,> 
(T,--g^Jlaw)= ela».> ■ 

where, for example, l^(i>^ denotes the atomic wave 
fTinction for proton CL and electron | . 

With the above wave functions it is possible to 
construct symmetric and antisymmetric wave functions for 

the system. 



t i«tt « i ia#' . r aifc »r » 



31 






The functions \C(y and lb) are not orthogonal. 
Define the overlap of these functions as Ls<l<i3[|bX 
With this and the assumption that the atomic wave func- 
tions are normalized it is possible to obtain the fol- 
lowing. 



iJ> = l+L\ <AIA)= l~L' 



<JIA> = 



,-<-. 



If the spin is considered it can be seen that there will 
be one antisymmetric spin wave function and three sym- 
metric spin wave functions. 

\ 



ia> = vT ^ >!.(+) Ha '-^■~n.^>n.uO 



where s'] is the spin function. The subscript identifies 
the electron and + or - denotes spin up or spin down. The 
simultaneous wave function must be antisymmetric. There 
are then the singlet state, Ua]> , and the triplet 
state lA^)^. The singlet state has spins paired and 
therefore no net magnetic moment. The triplet state is in 



32 

every way identical to a spin one particle with 
Mz= 1, O, -I 

By forming <^lHl^"> it is possible to obtain 
the energy shift for the singlet state 

where 
and 

T / \ \ Q" L. ^^ ^^ ^"^ 1 L_ /- \ 

K. represents the total electrostatic energy of the two 
atoms and J*, is the exchange integral. From <.A\H\A'> 
it follows that the energy shift for the triplet state is 

Next, consider the energy difference between the singlet 
and the triplet state. 

E, - E. = A E = ^'fl^K^'^ = 2 ^ 

If 0>Cthen the triplet state is energetically stable 

and the molecule will be magnetic. 



33 



Lei 



t-c — ^€.+ )— L** 



Using this, it is possible to write 

The total spin is a constant of the motion. 

5, and Si are also constants of the motion with 
eigenvalues -|- h . For the singlet state it can be 

shown that 

and for the triplet state one obtains 

Consider the spin Hamiltonian, 

It can be seen that it has the same eigenvalues as the 
electrostatic Hamiltonian used in the original formulation 
of the problem. This gives a spin-spin interaction with 



^■W^fltrM^anil' I' l l B Mtifc Tji i a n M w^ i nn M l> MiW** ln' M L' ''t:»' " >> 3 'wi.' *^ »i»<»>^ ^i>i 



34 

a weight factor Q that arises from electrostatic forces 

and symmetry requirements. This part of the Hamlltonian 
is called the Dirac-Heisenberg Hamiltonian, 

H= -2fl,,S."§;- (2.13) 



One can now give an approximate connection 
between the exchange integral and the Weiss constant. ^-5 
The assumption is often made that Ci-O for all atoms 
except the nearest neighbors and that Q - J'e ^o^ all 
neighboring pairs. Based on this 



Wex ^-2geY^SrS, 



where -^ indicates that the sum is to be taken only over 
nearest neighbors. Assume that the instantaneous values 
of the neighboring spins can be replaced by their time 
averages. Then 

where Z is the number of nearest neighbors. If the 
magnetization is along the z-axis then <S;<i'> - < Sy,-/^ =• (^7 

and 

This energy should equal the potential energy, V , of 
the spin l in the Weiss field ^ M . 



—^-^^...^.l: ---^.^ ...-■,■.-.■ ^,,^—11 ^.>1^^_-^^ -^. _■--- .- . 



35 



V= - ^Mt^SzijJii (2.15) 



Thus 



Using this and equation 2.9 it is possible to write 



T, = 2^g|S^^-'^ (2.16) 



Neel theoTj of antif erromapinetisni 

The Heisenberg theory of ferromagnetism is 
based upon the assumption that 0>O. When 0<^ an 
antiparallel arrangement of spins is favored and an anti- 
ferromagnetic substance is obtained. This type of system 
was' investigated by Neel,59 Bitter, ^ and Van Vleck, -^ 
and their work forms the foundation for the theory of 
antiferromagnetism. 

Consider two interpenetrating lattices made up 
of sites A, with plus spins, and B, with minus spins. 
Assume that there are antiferromagnetlc AA, AB, and BB 
interactions. Call these interactions w^g^, w^i^ and wi-,-]-, 
respectively. Since A and B are symmetrical w^a=^b= o( 
and Wa^T3=wt)a= & • The effective fields can then be written 



•t^tiffrnfaromm^ietmi^tM^* -it r 



36 



He«a = H - 0( Ma-(3 Mb (2.17) 

He-ff b -H-(3Mo--aMb (2.18) 

where H is the applied field and (X and e> are positive 
Weiss constants. 

Following the same methods used in the Weiss 
theory one can write that In the limit of high temperature 
and small X 

M^^MidJinA H^„^ (2.19) 

where N is the number of A atoms per unit volume. Simi- 
larily, if the dipoles on B are identical to the ones on A 
then 

3 k r 

Use equations 2.17-2.20 to obtain 

M = ^3>;iy^" [2 H -(o. .(3) ka] 

This becomes a scalar equation with the assumption that M 
and H are in the same direction. 

Y^ M __ 2NQyaJ{j^l)/3k _ C ^g 21) 



-'^''W-1 — rT|iiii'«-"lr't"^?i- r-T i ' i«i T .r»r rT «M - i» ■ f » <' i r n ~ i n ii " i r i - |. -) iir 'i j i ri ji n' «i i-M uiiii iw i» i iniia ■ m wii n i -^n^ iw ii la^ r ii r i w iiif rr-n- ■ ';Trtn- n un mi 



37 

This is quite similar to the result obtained for the Weiss 
formulation of magnetism. 

Next, examine the behavior at T~ Th' This 
temperature is still far enough away from saturation to 
use equations 2.19 and 2.20. With U~ write 

y" Ma = -J^UWio+^W\b>) (2.22) 

and 

JVib = -^^^ {(3Ma+(xM,J (2.23) 

where u is the magnetic moment per atom, U = qVa J"CT+1). 
From these it follows that 

T^J = -f^ i^-Oi) (2.24) 

} ■ Observe that Tw Increases as the interaction AB Increases 
and decreases as AA and BB Increases. A relationship can 
. now be established between Tn and Q by using equations 

I 

2.21 and 2.24. 

Jj± _ (^-^ (2.25) 

Experimentally it is found that Tiv/ < <9 which implies that 
j 0(> O or that, indeed, there is an antif erromagnetic AA 

,) and BB interaction. • 



/-'■> 

^ ..■■ 



38 

Phenomenolop:lcal discussion of orderlnp: in heavy rare 
earths 

Equations describing the types of ordering 

shown in Figure 2 may be written in the following form, 

|J* = gju^ JMCOS (5' Rn) (2.26) 



jUa^ = gjJ^ JMsin(^. RJ (2.27) 



|J* = gp,3 J M' Sin(^-Rn + g) (2.28) 

where the 2 -axis is taken along the crystallographic c- 

X H ^ 

axis and jJ^ , jU„ , jj^ are the components of the 
moments on an atom at f?n. M is, in this case, the 
relative saturation along both the X and the w axes 
and M is the relative saturation along the 2 -axis. 
The vector Ci is parallel to the o-axis and has a 
magnitude, q^—Zn/cd , where d gives the period of the 
magnetic structure. 

Equations 2.26 and 2.27, taken together, des- 
cribe a helical structure while equation 2.28 alone 
describes a longitudinal wave structure. More compli- 
cated structures occur and may be described by variations 
of the above equations. 

Next, examine the results obtained from a 
Heisenberg-Dirac form for the Hamiltonian. 



•t.'V tfJii " i~i-rc"- » - -1 1 ■ 



39 



/ 



^ 



H = -2^(R.-R^) Sn-S^ (2.29) 



since this exchange energy is, for the rare earths, 
usually much smaller than the splitting of the J" 



\ „y multlplets by the spln-orblt coupling, De Gennes"^ has 
proposed that S for each atom must be projected on the 
total momentum J . 

S=(3-')J (2.30) 

This comes from a phenomenologlcal approach and has been 
examined and shown to be valid by several workers. °3»o^ 
Using this expression and the above Hamlltonian one can 
-^ obtain the exchange energy for the helical ordering 
described by equations 2.26 and 2.2?. 

Esx=-2^(|)N(9-I)VM' (2.31) 

where 

^(?)=^P,«.-«„)Cos[^-(R.-R-o^] (2-32) 

For the longitudinal wave the exchange energy is 



2) Ze.^-^(i) NLg-\?J'U 



(2.33) 



where N is the number of atoms in the crystal. 

Note that these structures are energetically 
most stable at that "a which makes vJ(|) a maximum. Also, 
the spiral state is energetically more sta'ble than the 



n4ga-<^-~'1MBJlT^it~ii.-*T7fcMiT 



■jn^ n.ti!if., , 



^0 

longitudinal wave due to the factor of 2 found in the 
exchange energy of the former. However, in considering 
stable arrangements it is necessary to look at the free 
energy, F= U-JS. A molecular field approximation gives 
the same transition temperature for both structures. -5 »°°»°7 
This transition temperature can be written as 
Th= 2 Q[q)i^-if J iJ+\)/3k . One must look to the 
anisotropy energy to determine the relative stability of 
the structures, 

Anisotropy 

The term magnetic anisotropy refers to the 
dependence of the internal energy of a crystal on the 
direction of the spontaneous magnetization. The energy 
associated with this directional dependence is called the 
magnetic anisotropy energy. The dominant source of aniso- 
tropy in the rare earths is the electrostatic interaction 
between the multipole moments of the ^f electrons and the 
crystalline electric field. The crystalline potential 
for a hexagonal close pack structure takes the form ° 

where A^ are the constants determined by the distribution 
of charges around the ions and 2 is taken along the 
c - axis. The summation is over the coordinates of all 



•Ma#«3BS^»^ 



J 






41 

of the electrons. This can be transformed into a more 
convenient form by use of the Wigner-Eckert theorem, ^ 

+ pA:<r>[3Sj;-30j/ J(j+0+3J^(J+l)V25J.-6J(J+l)] 
+jA;<r>b3lj/-3l5j(jtl)j2+lo5jYjfifj;-5J'(J^0^ (2.35) 



where o( and s are constants which have been evaluated by 
Stevens, The < K"j^ are the mean values of \'" over the ^-f 
electron distribution and may be computed."^*-* The An 
are very difficult to evaluate and only order of magni- 
tude estimates have been obtained.'-'- 

If Ha is treated as a perturbation on Nay , 
it is found that at high temperatures the first term is 
the dominant one, but at low temperatures the higher 
order terms also become important. The first term 
corresponds to the quadrupole moment and causes the pre- 
ferred direction of the ordered moment to be either _!. 
or I! to the c-axls depending upon whether c<. is positive 
or negative. The second and third terms cause the 
moments to tend to align parallel to c when they are nega- 
tive, but when they are positive the preferred direction 
is at an angle from c . 






^n-^ 



-■^^ 



i 



i^2 

For dysprosium and terbium the first term is 
dominant and is negative over the whole ordered range. 
The ferromagnetic transition in these elements is caused 
by an increase in the fourth term with decreasing tempera- 
ture. For erbium the third term is positive and fairly 
large and makes the conical structure stable at low 
temperatures. 

This method of combining an exchange inter- 
action with crystal anisotropy has given very good quali- 
tative results. As yet no quantative calculations have 
been made due to the extreme complexity of the problem. 

Ranp:e of exchange interaction 

In order to obtain some idea of the range of the 
exchange interaction necessary to stabilize the screw 
structure we look at a particular model. "^^ Assume that 
the exchange interaction, , between layers of atoms 
perpendicular to the c-axis extends as far as second- 
neighbor layers. The exchange Hamiltonian now takes the 
form 



He. = -II 2f S,-S.-., 



L n = Q,±i.+-^ 



where Si, is the average spin of an atom in the ith layer. 
By summing this exchange Hamiltonian it is possible to 
obtain an expression for the exchange energy. Ey 
referring back to equation 2.3I it is possible to see that 



r'^*^ 






^3 
the assumed spiral configuration will be made most stable 
by the values of ^ that maximize H^^j . The 0(3) for 
this model can be written 

^(|) = ^0 + 2 g.cos^ + 2g^cosc^c , 

The value of O which maximizes this expression 



is 



S^ - _ Ji. . 



COS 2 - 4J, 

An analysis of the available data for dysprosium has been 
made by Enz"^^ and the values Ot/k —-Z^, 0, /k = 44 and 
Qi/k^ ~IS obtained. Similar results were obtained from 
an analysis of data on erbium. Observe that 0((i) is 
rapidly oscillating and long ranged to produce this spiral 
structure. Since the overlap of the ^f electrons on 
neighboring atoms must be quite small it would seem that 
this long range interaction is due to some other effect. 
It is reasonable to consider that the main part of the 
exchange interaction is produced by the exchange coupling 
between the conduction electrons and the localized spins. 
i 

Indirect exchange 
j_j Indirect exchange has been extensively investi- 

gated;'^^* 7^. 75 the following discussion closely follows 
that of Liu.'''" He starts by considering one conduction 
electron interacting with the magnetic electrons of one 
ion. The interaction Hamiltonian can be written as 



hrs~r^' Si r«S aii» S£^n ■-- *■ » i jT i i^ ^ ' ■ *" »■ • i IAnimti ^ n tii » r i wy ^m <* hi ' ••mn »i. .iMt^^M^i^ w i crfi i T ^ i ■^.■ ■^ ■■^■^ A^ g * 



.) 



) 



Itk 



L =1 

where Tn+i is the position of the conduction electron 
and r.- is the position of the ith magnetic electron. 
The wave function for the conduction electron is of the 
form 

T(r,s) = jUkC?) expl^k-r] rj (2.37) 

where 

is a Block function and T^ is the Pauli spin function. 
Since the 4f electrons are well shielded their wave 
function can be written in the form 

^ipCn^ = ^mY,^(o,^) ri (2.38) 

The wave function for the entire shell is constructed 
from the single particle wave function as prescribed by 
Hund's rules and the Pauli exclusion principle. Since 
this dissertation deals only with dysprosium and erbium, 
for i^hich the ^■f shell is over half full, only that case 
will be considered. 

Liu^ ° shoxTOd that the required wave function of 
the shell is 

^irA~Y^ C(LSJ; m, M-m^ A«t'YL^,tT5,NN-m,t (2.39) 



— <wii«a>*im(fr^l«».i>qi»ar% J 



i^5 

x-rhere C(LSJ;hfijVl-nr> ) denotes the vector coupling coef- 
ficient and the sunmation over t refers to a surrmiation 
over Young's diagrams.'' 

The vxave function of one conduction electron 
and one magnetic shell with no regard to symmetry is 

^= Yxm(i,...,n)^(n^.O (2.40) 

where "^nih) ~ "^^t^Kvi ,sn+i)» Next, this wave function must be 
antisymmetrized with respect to all the N+l particles. 
The resulting wave function is 

N 
(N+-0^» L'Yj-/v\Cl,-,N')'^(lvH-0— ^ Tj/v\(l,...,t-i,Nt-i, ii.i^...N")'^i.y (2.4l) 

The particles are considered to be completely Indis- 
tinguishable; therefore equation 2.36 must be symmetrized. 

u ~7 _^^\^ (2.i^2) 

where i->i^^<N + l and i^^-^' . Consider the following unsym- 
metrized initial and final states. 

'^f = "^XNv'(i...,N)^tN4.) (2.^^) 



where 






■ r- «i 'ifir ri" • '^ r~ w n irftf ii ' ' ■ ■ i m » ' nm w n i rf « > 1 wii — | - 



if6 

Using the expression for Tjtm , equation 2.39. one can 
now form the matrix elements of the exchange interaction, 

Liu obtains an expression for Hx for the 
heavy rare earths which includes direct interaction 
^ between shell electrons, exchange interaction between 

shell electrons, direct Interaction between conduction 
and shell electrons and exchange interaction between 
conduction and shell electrons. The last of these is 
found to be 

i 

H = - 2 I(M')(g-l) S'J (2.^5) 

Jjj- where ICk.k'") is the exchange integral, S is the spin 
of the conduction electron and J is total angular 
momentum of the ion. In order to obtain equation 2. '4-5 
Liu made the following approximations. 1) The conduction 
electrons are s electrons so their wave functions have 
spherical symmetry. 2) The wavelength of the conduction 
electron is large compared with the size of the i^f shell 
so that eKp^ik'l}) may be approximated by the leading term 
of its power series expansion. 

It is very difficult to justify the first approxi- 
mation and Liu did not try to show that it held. The 
second one can be examined by looking at the radius of the 
^f shell as determined by the method of Pauling. 78 it is 



5" 



I 



^7 
fo\ind to be about 0.4 A. Using the free electron approxi- 
mation it is found that, for the heavy rare earths, 
k~ \.S X 10^ cm"-'- at the Fermi energy. Therefore, 
k '"F;»0.6,and the second approximation is seen to be reason- 
able. 

Recently Kaplan and Lyons'''3 have examined this 
second approximation and found that the leading term does 
indeed dominate for terbium through erbium and that the 
correction by other terms is about 10 per cent. 

De Gennes°2 has found that since I(k,k') should 
be the same for all rare earths Tc or Tw should be 
proportional to (^-i)^J ( vT+l). For the heavy rare earths 
this reduces to S'CJ'+O/J. This is the same result that 
Neel obtained in 1938"''9 based on the molecular field 
approximation. This relationship is verified experimen- 
tally except for ytterbium. 

Pressure effects 

Using an equation first derived by Neel, 
Robinson et al.39 have constructed an interaction curve 
for the rare earths in an attempt to predict the effect 
of pressure on the transition temperatures. The Neel 
equation is 

J± ^ ^^g^j (2.46) 

k 2iiS'(J'+0 

where (^ is the transition temperature and 2 is the 
number of nearest neighbors. Using known values of the 



-- ' " ■ T " " -——'—>—— ——T~ 



k8 
right hand side of this equation the quantity Oe/k was 
calculated and plotted versus D/2R where D is the inter- 
atomic spacing and R is the radius of the ^f shell. 
There is some difficulty in using &^ in equation 2.^6 
for the rare earths. In materials which go directly 
from' the paramagnetic to the ferromagnetic state Oct , 
the temperature at which the material actually becomes 
ferromagnetic, is a few per cent lower than Ocp , the 
extrapolated transition temperature, so it makes little 
difference which is used. In the heavy rare earths where 
antiferrom.agnetic states are observed these two tempera- 
tures are far apart. To account for this, two curves 
were plotted, one based on Od and one on Ocp , The 
resulting curve is shown in Figure 22. ^ This analysis 
accounts for the magnitude and sign of the dTc/d P found 
for gadolinium and terbium and can be used to explain their 
result that dTc/^Pfor dysprosivim is positive for low 
pressures and changes to negative as the pressure is 
increased. 

Liu has done an analysis of the effect of pres- 
sure on Tc for ferromagnetic materials. ^0 He has looked 
specifically at gadolinium but was able to draw conclusions 
about the behavior of heavy rare earths from his work. 
The starting point chosen for this analysis is the Indirect 
exchange Hamiltonian 



^9 
where the first term is the kinetic energy of the conduc- 
tion electrons in the scalar effective mass approximation 
and the second is the exchange interaction between the 
conduction electrons and the ions. The subscript i 
refers to the conduction electrons while /j refers to the 
ions. As in the previous discussion of the indirect 
exchange the electron is described by a Block function. 
The matrix elements of equation 2,4? are written as 

It has been shown^^ » "^^ » "^-^ that by second order pertur- 
bation theory the exchange interaction can be expressed 
by the spin Hamiltonian 

With 

^^^^"' In- C2^p-n-R- [2k.RC05{2krR)-Sin(2i<,R)] (2.^9) 

Where 1 is the average matrix element for k and k 
approximately equal to k^ . Equation 2.^9 can alternately 
be xwitten as 

JCr^l- ^ttrN(e,)F(2,,R) ' (2.50) 



men J ^m »i—i m^ — i ,tmmm rtr^nK- ■ T » iTTWfe . -- : a» c iej -- ■ ; 









50 

where H is the valence of the ion, N(£f) is the density 
of states at the Fermi level and F^) is given by 

De Gennes"'^ has shown that it is reasonable to assume that 
the ferromagnetic state is the ground state for gadolinium. 
Following this we can write 

Ey= fill's* N(£.)^ F(2k.Ri) (2.51) 

This equation should now be examined for terms which will 
vary with pressure. The summation will be independent 
of volume if the electron distribution is isotropic. The 
2 and S are independent of volume in the heavy rare 
earths. One can now take the logarithm of equation 2,51 



^ and form 



ainE __ Dlnlll" Din N(£f) /g 52^ 

■^\r^y ~ 7)ln\/ "^ 2>|n V ~ ' 

The Curie temperature is proportional to the ordering 
energy per spin, therefore we may write 



0\r\Tc. DIolTI' , Din N/ce/r) /« ^^^ 

DInV ~ ainV DInV ^^oj; 



The terms on the right side of equation 2.53 are unknown 
at the present time* Liu gives some estimates of the 
limits that can be expected for them. The thing that 
should be noted about them is that they are both functions 
only of the electronic properties of the material. Since 



■1 iiM ■•««- - ^-r~ 



J 



51 
all of the heavy rare earths have similar electronic 
properties, gj|^ ^^ should be the same for each. This is 
a rather strong assumption and should be sub;5ected to 
experimental verification. This is discussed further in 
- Chapter IV. , 

Thermodynamics of phase transitions 

It is generally accepted that the transition 
from the paramagnetic to the antiferromagnetic state is 
second order while the transition from the antiferromagnetic 
state to the ferromagnetic state is first order. 52 

It is possible to characterize a first order 
phase transition by either of the following statements. ^^ 
i; There are changes of entropy and volume. 2. The 
first order derivatives of the Gibbs function change dis- 
continuously. Any phase transition that satisfies these 
requirements is known as a phase change of the first order. 

The effect of pressure on a first order phase 
transition can be determined simply by taking the first 
TdS equation of thermodynamics and integrating it over 
the change of phase. The first Tds equation can be 
written, 



Tds = c^dT -t-Y^j du- (2.5^) 



52 

Integrating this it is possible that one obtains, 

dT --^ 



^-^ . (2.55) 



In this equation the superscript f refers to the final 
phase and l refers to the initial phase. 

A second order phase transition is charac- 
terized by discontinuous changes in the second order 
derivatives of the Gibbs function. There is no change 
in entropy associated with this transition. Using the 
same superscript notation as before it is possible to 
write S'= S^ at (T, P) and S'-^ds = s'+-as^ at 
Cr+dT, P + dP) , These expressions yield 

Tds' = Tds' (2.56) 

The second TdS equation is now used, 



Tds=c,dT-(^j_dP 



By using equation 2.56 and the definition of the volume 
expansivity 

it is possible to write 

CpdT-Tu-p'dP - oldT' Tu-(3^dP 

3y re-arranging and using the relation B = 3oi and P= — 
one obtains 



)■ 



53 



_dT _ yr _A^ (2.57) 

where A(X-o(^-a:' and AC^ct-C^, . Equation 2.5? is 

knoim as an Ehrenfest equation. This equation predicts 
the pressure shift for the Neel transition and is further 
discussed in Chapter IV. 



\ 

J 



) 



CHAPTER III 
APPARATUS A1^"D PROCEDURE 

Introduotlon 

A description of the apparatus and of the pro- 
cedure involved in the measurements made in this disser- 
tation can be roughly divided into four major sections. 
The first section is concerned with the detection of the 
ferromagnetic-antiferromagnetic and antif erromagnetic- 
paramagnetic phase transitions. This task is complicated 
by the fact that the sample is contained inside a pressure 
bomb which is in turn contained within a temperature 
control cryostat. Further complications arise from the 
safety requirement that every thing should be operated 
remotely. The second section deals with the techniques 
involved in the compressions containment and pressure 
measurement of helium gas at high pressures and low temp- 
eratures. The third section concerns the production and 
measurement of temperatures from 5°K to 190°K, and the 
last section gives a step-by-step breakdown of the procedure 
used in performing the experiments. 



5^ 



55 

Detection of Ma,g;r.etlc Transitions 

Several methods have been used to detect magnetic 
transitions In rare earths. The most Important of these 
are neutron diffraction, bulk magnetic measurements, 
specific heat measurements and electron .transport property 

^ measurements. 

It was decided to look at the bulk magnetic 
properties in these experiments since they promised to give 
sensitive indications of the transitions, would readily 
lend themselves to pressure studies, and did not require 
any elaborate instrumentation. Methods of detection of 
the transitions by bulk magnetic properties are mentioned 
in Chapter I. Several factors had to be considered in 

^''"' deciding upon the proper method to be used. It was 

desired to have as much sensitivity as possible; therefore 
a bridge method was selected. A large filling factor was 
desirable so the coil was placed inside of the bomb. 

Since the working space was limited and since the number 

1 

1 of electrical leads into the high- pressure region should 

1 be minimized, it was decided to use a single coil tech- 
nique. After the experiments were well under way Samara 
and Giardinl37 reported that they had used the same 
method. The sample, in the form of a cylinder, was placed 
within a solenoid and the self -inductance of the coll was 
monitored. There is no simple exact formula for the self- 
inductance of a solenoldal coil of practical dimensions. 







5^ 

An approximate formula is 



60. +^b4 iOc 

where CI is the mean radius » b is the length, C is the 
radial thickness of the solenoid and n is the number of 
turns. The important thing to note is that L is pro- 
portional to /J , the permeability of the core material. 
As a ferromagnetic sample is heated through its Curie 
temperature, its permeability changes from a large to a 
fairly small number. Hence, if the inductance of the coil 
is monitored, a large drop is seen as the sample is heated 
through its Curie temperature. The transition from the 
antiferromagnetic to the paramagnetic state is accompanied 
by a peak in the permeability versus temperature curve. 
Of course, the inductance of the coil would also be a ■ 
function of the thermal expansion of the copper wire and 
the sample, and of variations of jJ due to skin effects. 
A blank run was made to insure that the changes in the 
coil were not influencing the results. Since the magnetic 
changes are quite large it was reasonable to neglect the 
other effects. 37 

Inductance bridge 

In order to perform these experiments a very 
sensitive self-inductance bridge was needed. The design 
of inductance and capacitance bridges has been advanced 
considerably in recent years with the development of very 



57 

accurate ratio transformers. "^toit- These instruments 

utilize modern high permeability magnetic core materials 
and are highly accurate alternating voltage dividers. ^5 
A ratio transformer bridge was built following a design 
by Hillhouse and Kline. °° This bridge was capable of 
detecting changes of Inductance of the sample coll of one 
part per million. 

The wiring diagram, Figure 5, shows the com- 
ponents as connected in the bridge. This design features 
the use of commercially available components as listed 
below: 

1. Audio oscillator, Hewlett-Packard Model 200JR 

2. Isolation transformer, Gertsch Model ratio 4-1 

3. Ratio transformer, Gertsch Model 1011 

if-. Decade resistance box. General Radio Type 1432-K 

5. Null detector, General Radio 1232-A 

6. Standard inductor. General Radio 1482-L 

7. Standard inductor, General Radio 1482-H 

All of these components, except the null detector and the 
audio oscillator, are contained within one cabinet. 
Figure 6. All of the external wiring is coaxial cable. 
(GR 874-R34) with General Radio shielded connectors. The 
switch, S-j^, allows the isolation transformer to be connected 
with a ratio of 4:1 or 1:4. The 60O ohm generator output 
impedence ceui then be transformed to 37 ohms or 9OOO ohms. 
The purpose for this approximate impedance matching was to 



58 



&fv® 



OQOQ 




I ^-S* ^- 

OOO0 ^OQ 



-TV- ^ 

■■0 ' 6 



Ratio 

Transformer 



i^— ■— ^^^— ^^ 



Unknown 
Inductor 




Standard 
" lOOma 

©- 





Decade 
Resistor 



) ' & 



-©—6 



M' Detector 



Figure 5. Wiring diagram for the inductance bridge. 



c 



m 

o 

OS 

§ 

■H 



0) 

I 

•H 



-^: 



I 



61 
realize good bridge sensitivity. The ideal ratio between 
unknown inductance and standard inductance is 1:1; however, 
Hillhouse and Kline found that the accuracy was not 
appreciably altered up to a ratio of 10:1. Switch 33 
allows for the use of either the 10 mh or the 100 mh 
standard inductor. The equivalent circuit for the bridge 
is shown in Figure ?• The operating equations for the 
bridge are derived below. Standard notation is used with 
subscripts 1 and 2 referring to the leads running from 
the bridge to the sample coil, S, to the standard, D, to 
the decade resistance and X, to the sample coil. The 
reading on the ratio transformer 1 A , is that part of the 
total voltage that is being applied across the unknown 
inductor. Looking at the schematic it is seen that at the 
balanced condition, that is, when the current through the 
detector equals zero, one can write 

e,= E(l-A')=l[R.-fRs + Ro+^u;(L, + Ls+Lp)] (3.1) 

ez=EA= I[Rx + R2+^u;iLx+L^)] (3.2) 

Dividing equation 3.I by equation 3.2 gives 

(I-A[Rx+Ri+<ju;(Lx+U)J = A[R.+Rs + Rd+^u;(L, + Ls + U)) (3-3) 
Thus 



R>c=j^ fRs + Ro + RO-Ra. (3.^) 



62 

and 

Lx= j:r^{Ls + LD + L.)-Lz (3.5) 

Equations 3.4- and 3.5 constitute the operating equations 

for the "bridge. 

J 

^ The inductance of the decade resistor is given 

by the manufacturer and at the maximum is on the order of 
a |jH. The inductance of leads 1 and 2 are also on the 
order of a juH. During an experimental run the tran- 
sitions occur over a small temperature range; therefore 
Lp ,1.. and La are small and essentially constant. 
The Lx is from 1/2 to 1 Henry and changes in it completely 
dominate the picture. The situation with the resistances 
f^ is similar. Ri and Ri are small and essentially constant 

during the determination of the transition temperature. 
Using this information, one can truncate the operating 
equations and simplify data reduction. The simplified 
equations are: 



Rx = -j3a(Rs + Rd) (3.6) 



J 



L^=- -pf^ Ls (3.7) 



Hillhouse and Kline have made a detailed error analysis 
for this bridge design and found that it is able to inter- 
compare inductances at ratios as large as 10:1 to 



r irt'irii- r-rmr" 



63 



L R. 



J 



) 




L; 



R. 



Figure ?. Equivalent circuit for the inductance bridge. 



64 



accuracies an order of magnitude better than the certifi- 
cation limits of present standards which is at best t.OJ 
per cent. 



J 



Coil and sample description 

The two coils used in these experiments were 
wound on a teflon core with a Model W coil winder manu- 
factured by the Coil Winding Equipment Company. The 
dimensions and room temperature characteristics are: 



Table 3. Sample coils. 



/ 



Number of turns 

Length 

Inside diameter 

Outside diameter 

Resistance 

Inductance 



Coil 1 

14,000 
13/16 in. 
1/8 in. 
7/16 in. 
3100 ohms 
400 mH 



Coll 2 

17,000 
13/16 in. 
1/8 in. 
1/2 in. 
6000 ohms 
500 mH 



The samples were obtained from Leytess Metal and 
Chemical Corporation who specified a purity of 99.9 per cent. 
When received they were in the form of rods 6 in. long and 
.375 in. diameter. These were cut and turned down to a 
final sample size of 1/8 in. diameter by 13/I6 in. long. 
The samples were not annealed after machining. 



'~~i7-T~~~rViM*'i'r rt 'ntr~^ — — .— 



^5 

Pressure Generation and Measurement 

The purpose of these experiments was to study 
the effect of hydrostatic pressure on magnetic transitions 
in rare earths. A large high pressure helium gas facility 
V7as constructed to achieve purely hydrostatic pressure 
over most of the temperature range covered. In principle, 
it is easy to achieve hydrostatic pressure in the fluid or 
gaseous phase of helium. In the lower temperature region 
approximately hydrostatic pressures may be achieved by 
applying the desired pressure to the helium while it is in 
the fluid phase and then freezing it at constant pressure. 
Further cooling necessitates the calculation of the 
pressure from the equation of state of solid helium and 

J the thermal properties of the high-pressure bomb,^'' which 

was made of beryllium-copper. This procedure gives very 
nearly hydrostatic pressure even though there is some 
movement due to the fact that helium has a larger thermal 
expansion coefficient than beryllium-copper. 

Numerous experimental difficulties arose during 
the course of the experiments. By far the largest problem 
was leaks in the pressure system. The bomb plug seals 

J. presented the most difficulty since a leak there made 

temperature determination and control impossible. Cooling 
through the freezing temperature of helium had to be done 
very carefully to prevent blocking of the inlet pressure 
line before the helium in the bomb was completely solidified. 



66 

This would have greatly reduced the pressure at the sample 
as well as the accuracy with which it was known. 

Hip;h-pressure room 

The safest way in which to conduct high pressure 
J experiments is not to have any personnel in the vicinity 

■ of the high-pressure equipment. This was done by isolating 
all of the high-pressure components in a specially con- 
structed, explosion proof room below ground level. This 
room was located outside the basement of the low temperature 
laboratory. All of the pressure equipment plus cryogenic 
apparatus was operated remotely from the adjoining base- 
ment. A brief description of this room V7ill now be given. 
Figure 8 shows an outside view of the room. The 
wall on the left is the outside wall of the Physics building 
basement. Figure 9 shovjs a top view of the room and part 
of the laboratory, giving wall details and rough dimensions. 
The roof of the room was constructed, from inside out, of 
1/2 inch aluminum plate, k inches of sand, 9 inches of 
reinforced concrete, a ^ inch air gap, 1/2 inch plywood 
sheet, and a layer of sand bags resting on this plywood. 
Over this was placed another 1/2 inch plywood sheet which 
was covered with roll roofing. The outside end of the air 
gap was covered with screen and provided ventilation as well 
as a path for escaping gas in the event of an accident. 
The free volume of the compressed gas was only about one- 



r 



-f" 



4 



i 

O 

m 

0) 

h 
i 

-ri 

o 

o 

i 

-P 

a 



,# 

v 



CO 
(D 

to 



69 



.; 



A_,/ 










gfea^;^^s;:^:^:-:g^:e?a^^-g^°?gQ-§>l^ 



~CS 



;■/ :^" 






,-y-,-^^:^.^c?;-c?-.<?,.-^; 



,0 o r J. 




•H 
O 



H 
t-i 
O 



a 

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u 



H I 

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o ft 





o 
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u 

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ft 
I 

fcO 

o 




o 



ON 

u 

fao 

•H 



70 

fortieth of the volume of the room and therefore could 
not create a significant increase in the pressure of the 
room. However, liquid nitrogen and helium dewars as well 
as the commercial helium gas tank were left in the room 
for remote transfer and if one of these should be rup- 
] tured by shrapnel it would release enough gas to be 

dangerous in a poorly vented room. The door was a 5 1/2 
inch thick box made of -1/2 inch alximinum plate and filled 
with sand. The box was supported by a steel dolly which 
had 6 ball bearing steel wheels that rolled in the channel 
of a 6 inch steel I beam. The north and east walls of the 
room (Figure 8) consist of 12 inches of steel reinforced 
concrete backed by earth. The south wall was constructed, 
from Inside out, of 1/2 inch aluminum plate, 6 inches of 
sand, and a wall of 8 inch solid concrete blocks. The ' 
west wall consists of the outside wall of the Physics 
building, 15 inches of reinforced concrete, supplemented 
by a 1/2 inch aluminum plate and 6 inches of sand. It 
was deemed necessary to add this plate and sand to prevent 
Spalding of the concrete wall in the event shrapnel struck 
the wall.^S The room was designed to contain all shrapnel 
and shock waves in the event of a high-pressure gas failure. 

Kip;h"pressure gas apparatus 

The high-pressure system is a three stage system 
composed of an Aminco 30,000 psi (H 5968) oil-to-gas 



J 



/■ "^ 



71 

separator, a Harwood 100,000 psi Intenslfier (SAlO-8- 

1.250-lOOK), and a Harwood 200,000 psi intenslfier 
(SAIO-6-.875-2OOK). Figure 10 is a schematic showing 
all of the significant coEponents. Figure 11 shows the 
relative size and the placement of the components within 



'j the room. 



Initial charging was accomplished by a remotely- 
operated solenoid valve (switch located on the panel). 
For safety, a second solenoid valve was used to bleed the 
2,000 psi stage of the gas system after charging. The 
charging gas flowed through a liquid nitrogen cold trap 
and a filter to remove gas and solid impurities. Mote 
that each stage was separated from its lower pressure 
adjoining stage by a one way ball check valve, as shown 
in Figure 10. The check valves in the 30,000 psi stage 
were Amine o No. ^^-6386 while the ones in the other stages 
were Harwood I4L-603. 

The Helse gauge. Model H 2696O, located in the 
100,000 psi stage, was monitored by a closed circuit tele- 
vision system which consists of a Marson television monitor 
and a Bell camera. This gauge proved very useful in 
locating leaks and controlling bleed down rates in the 
system. The pressure during an experiment was always 
measured with the Harwood manganin resistance cell. This 
pressure was monitored constantly by a Poxboro recorder 
but data points were taken with a Carey-Poster bridge. 



ii .» — ■ — »<^'— v-T-T»-^ ■ ■-- Ti ^^n mn m imii 



72 



) 




CO 

(>> 

CQ 

CO 
CQ 
<D 

I 

60 



O 

o 

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CJ 

o 

CO 



o 

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0) 



a 
o 
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u 

(0 
CO 

to 

u 
p< 
s 

cO 

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s 






... ^.--^ ^...^-^.i-^- ^ «iU«.. ,5v 



) 



) 




^ 



75 

Harwood Hajiufacturing Company specifies an error of less 

than 1 per cent with this cell and bridge. The schematic 
also shows the motor driven bleed valve. This was a 
Harwood 200,000 psi needle valve driven by an electric 
motor through a chain and sprocket arrangement. 

Control panel 

Figure 12 is a schematic of the control panel 
and Figure 13 is a photograph of the actual panel. The 
schematic shows that the control panel was divided into 
two separate pumping systems; a 30 » 000 psi oil system for 
the gas-to-oil separator, and a 2,000 psi oil system for 
the two intensif iers. Sprague Engineering Corporation 
air powered pumps were used for both systems. As a 
safety feature the air supply was taken through a 115-volt 
ac solenoid valve that was normally closed. In the event 
of electrical power failure, affecting other components 
of the facility, the air supply was automatically stopped 
and had to be manually tripped on when power was restored. 
The intensifier oil system operated at reasonably low 
pressure so it was practical to use one pumping system 
with electrically coupled solenoid valves to draw oil from 
the proper reservoir and to direct it to the proper inten- 
sifier. These solenoid valves were operated by push 
button svjitches on the panel, and pilot lights indicated 
which reservoir the oil was taken from and the intensifier 



OJ H 



1^ 



J 



U 

o 

E 

o 

Eh 




u\K 






/? 



en ft 



0) 

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d 

a 

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o 

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&< 
ft 



-P4 



.u 



^1 
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o u 




S 

0) 

-p 

CQ 
0) 



CO 
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G) 
?^ 
ft 

(D 

-p 

(h 
O 

<M 

iH 

ft 

H 

o 

-p 
s:: 
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0) 

o 



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plH 



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ft 

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79 

to which it was pumped. Each oil reservoir was fitted 
with a level indicating sight tube. These were calibrated 
to shov; the actual position of the appropriate piston in 
the high-pressure room. Stainless steel tube, type 30^, 
with 3/8 inch OD was used throughout the intensifier 
systems while 30*000 psi Aminco high-pressure tubes and 
fittings were used in the separator system. 

The photograph of the control panel (Figure 13) 
shows the position of the pumping controls. This panel 
was designed for maximum safety, efficiency, and conven- 
ience of operation. A color coded flow diagram was 
painted on the panel to clarify the oil and high-pressure 
gas circuits. Also shown on the panel is the Foxboro 
Dynalog recorder which- was used to monitor continuously 
the Harwood manganin resistance cell located in the 200,000 
psi gas stage. A Harwood Carey-Poster bridge (visible on 
the shelf in the lower right-hand corner of Figure 13)was 
used for accurate readings of this cell. Visual obser- 
vations in the room were made with closed circuit television. 

The push button and two warning lights in the 
upper left-hand corner of the control panel (Figure 13) 
remotely controlled the electric motor operated bleed valve 
in the 200,000 psi gas stage which bled this high pressure , 
back into the 30,000 psi gas stage. A safety microswitch 
was located at the 30,000 psi stage oil reservoir which 
prevented operation of this motor unless the separator was 



••|aiU-«:*«*r«wl «f r=wg*«.aMi>. '>»^ ■ - tt^ — i m g»r B-gtt-JM 



80 

I cycled all the way back, thereby providing adequate volume 

for the 200,000 psi gas. 

HiRh-pressure bomb 

In high- pressure, low-temperature experiments 

/■ ""■ 

J conflicting design considerations occur with the sample 

container (bomb). The bomb should be massive and strong 

to safely hold the pressure yet it must be small with low 

thermal mass so that it will fit a cryostat of reasonable 

size and can be cooled to low temperatures without using 

an excessive amount of liquid helium. Further, it was 

planned to use this bomb for magnetic measurements and 

nuclear resonance work, so a non-magnetic material was 

' - required. The desire for a non-magnetic bomb with high 

strength led to the choice of BeCu (Berylco 25) for the 

bomb material. 

A design pressure of 225,000 psi was used so 

that the bomb would be reasonably safe with the 200,000 

I 

psi gas system. Standard thick-v/all cylinder equations°9 
gave stresses exceeding the yield strength of full hard BeCu 
regardless of the x^rall ratio (outside diameter/inside diam- 
-. eter). These same equations, modified for a double wall 

cylinder, ^ predict sub yield point maxim-urn stress for the 
dimensions shown in Figure 1^. The outer cylinder was a 
.010-inch interference fit (on diameter) over the inner one. 
Assembly was accomplished by cooling the inner cylinder in 



81 



1" X 1^ KF X li" 



^.25" 



.65'' 



.Q75" 




ClTft =175.000 psi 



Fip^ure 1^. Righ-pressure gas bomb. 






«fW^U«>^t^<aA^.C.>*wtjb. ' «^u<>4juS^;' 



/'^A 



82 
liquid nitrogen and heating the outer cylinder to 900°P, 
then pressing them together. The assembly was then heat 
treated at 600°F for 3 hours. Final machining on the 
inside bore and sealing surfaces was done after the outer 
cylinder was fitted. Figure 1^ shows the finished bomb 
with dimensions and the tangential stress (7^ at the 
critical design points as calculated for an Internal 
pressure of 225.000 psi. It was very important that all 
corners and edges be made round and smooth to reduce 
stress concentration. 

Before use in the helium gas system, the bomb 
was pressurized with a liquid test system to 1^ kllobars 
and carefully checked and measured for distortion. 

The platinum resistance thermometer was installed 
on the bomb by means of a band on the outside of the bomb. 
This was assembled in the same manner as the bo., itself 
and was a .03 inch interference fit. 

Hl.g:h -pressure seals 

In performing these experiments one of the most 
difficult problems encountered was the design and fabri- 
cation of the electrical and bomb plug seals. The problem 
• of containing helium gas under pressure at low temperature 
is well known to anyone who has worked in this field. 
Epoxy seals, which were sufficient for much of the range 
covered, had already been developed at this laboratory. 90 



m 'f *, tttt ^^ mmm c t**'** 



83 

It was desired, however, to develop an electrical seal 

which would be easier to work with than this type. The 

electrical seal which was finally used is shown in 

Figure 15 and is simply a logical next step from the ones 

previously developed. The earlier seals depended upon 

y the bond betx-j-een the epoxy and the tube to carry the shear 

load vihich prevented the seal from blowing out. The new 

seal has a large cross section in the middle so that the 

epoxy itself must fail for the seal to blow out. 

In this seal, the wires pass through a small 

hole filled with an epoxy. (Sccobond 10^) The tubing 

used in the seal was Karwood 3^ a^d 12H. Standard Harwood 

cone and sleeve fittings which were good to pressures 

greater than 14- kilobars were used in all cases except 

where the 3M tubing mated to the 12H. One non-standard 

part, the gland nut, had to be made for this connection. 

It was made from type 30^■ stainless steel and had the 

dimensions shown in Figure 15- The 12H tubing was drilled, 

tapped and a 60° conical seat was made on each end. The 

3M tubing was threaded and coned in the standard manner. 

In order to insure that the epoxy bonded to the tubing, 

the walls were etched with acid and cleaned with water and 
t 
I " acetone. The wires used were number 36 quadruple formvar 

j insulated copper. They were cleaned with acetone and 

placed inside the tube. Epoxy was mixed and forced into 

the tubes with the small stainless steel tube and screw 



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o 

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85 

shown In Figure 15» The wires were moved slowly back 
and forth several times, the excess epoxy was removed and 
the seals were baked according to manufacturer's specifi- 
cations. The seals were tested on a liquid pressure sys- 
tem before being used In the gas facility. All of them 
were tested to about 1^1-0,000 psl and none blew out. Two 
of them were taken to 200,000 psl and while no leaks devel- 
oped the wires Inside were broken by the epoxy shifting in 
the tube. The addition of 5-10 per cent alvunlna powder to 
the epoxy has been reported^l to significantly Increase its 
strength. This was tried but the one seal made in this way 
leaked at a low pressure. This may have been due to incom- 
plete filling of the tube during fabrication. More work 
should be done on this since it seems that if the epoxy 
could be strengthened this should make a simple, inexpensive 
seal for gas systems up to 200,000 psl. 

The plug seal offered even more difficulty than 
the electrical seals. Many variations were tried and dis- 
carded. The one finally used is shown in Figure 16 and is 
good to at least 100,000 psl. It will probably go higher 
but leakage in the intenslfier seals have restricted the 
experiments to this pressure. It will be noted that it is 
a Bridgman unsupported area seal.° Some workers have 
reported that this seal will not work at low temperatures 
due to the fact that the indium metal contracts more than 
the beryllium copper upon cooling. It was found, however, 
that if the bomb was cooled while under pressure no leaks 



86 



1' 

2 

Y 



1^ 



V 



-V 



-Y 



1" hex head 
nut 



1-1^ NP 



1/8" 



annealed BeCu 

indi\«n 
hardened BeCu 



0,500- 




^^6o --^YA 



\ / 
\ / 

V 




4 i 
3/16" 



9/16" 



0.625" 



Figure 16. Bomb plug and seals. 



87 

would occur. This seal has the advantage that it seals 
with very lox^ torque, is simple to machine, and is 
reusable many times. It consists of a lower support 
ring made of hardened beryllium copper, a ring of indium, 
and an upper extrusion ring of fully annealed beryllitm 
) copper. The plug is made of hardened beryllium copper. 



Temnerature Production and Measurement 

The basic cryostat system, which was used in all 
of the experiments, is show-n in Figure 1?. Some modifi- 
cations, to be described later, were necessary for running 
the lowest temperature transition. The basic system 
consisted of an outer nitrogen dewar, an inner nitrogen or 
helixm dex^rar, and an evacuated can x-rhich contained the 
bomb. Aluminum foil, which has a low emissivity, was used 
to X"jrap the bomb and line the inner x-ralls of the can. The 
inlet pressure line was stainless steel which has a low 
thermal conductivity at low temperatures, A non- 
inductively would heater coil was connected to the top of 
the bomb with woods metal. Three copper-constantan 
thermocouples were installed to measure the temperature of 
the bomb. The upper and lower ones used a copper lug on 
the X'rall of the can as a reference temperature while the 
middle one was brought out of the cryostat to a liquid 



88 



Hip;h-pressure gas 

I 



Thermocouple and 
heater lear'*^ 



Helium transfer 
line 




Electrical 
seal 



>G<^ 



> Vacuum 
pujnp 



Inner dewar 



Outer dewar 



Heater 



Thermocouples 



Helium level 
resistors 



Figure 1?. Cryostat. 



y 



89 
nitrogen bath for reference. Periodically a platinum 
resistance thermometer was installed in the bomb and the 
thermocouples were checked. Temperature accuracy was 
judged to be -.25°K and reproductibility was better than 
±.10°K. 

In the experiments involving the Neel transi- 
tion the inner dewar was filled with liquid nitrogen and 
exchange gas was allowed into the can containing the bomb. 
V/hen the bomb had cooled to approximately the desired 
temperature the can was evacuated and thereafter main- 
tained at a pressure of not more than .2 microns. With 
the bomb thus isolated the required temperature could be 
maintained with the heater. 

In the experiment involving the Curie transition 
of dysprosium it was necessary to cool the sample down to 
about 60°K. This was done by pumping on the nitrogen bath 
with a large Kinney pump (Model No. KC-i^6) . The tempera- 
ture was then controlled by use of the heater. 

The ferromagnetic transition in erbium, vrhich 
occurs at about 20°K, made it necessary to use liquid 
helium and also required some modifications in the equip- 
ment. Two copper straps (dimensions 3 x .375 x .025 inches) 
were soldered to the bottom of the bomb and to the bottom 
of the can. In addition to this, a heater coil was wrapped 
around the inlet pressure line. With these two modifi- 
cations it was possible to maintain a temperature gradient 



90 
across the bomb sufficient to insure that the helium 
froze from the bottom to the top. 



frocedure 

Proced'are for experinents above 60°K. 

The following procedure is applicable for 
experiments above 77'^K» A modification of the procedure 
given at the end of this section allows for operation 
down to SO^li. 

1. Place the sample in the coil and solder the coil leads 
onto the electrical leads in the bomb. 

2. Clean the seals and install the plug in the bomb. 

3. Wrap the bomb with aluminum foil and place the styro- 
foam spacer around the bomb. 

^. Woods metal the can into place around the bomb. Check 
the system for leaks in the joints. 

5. Lower the can into the inner dewar and bolt the flange 
into place. 

6. Carry the dewar system into the pressure room and connect 
the pressure lines and all of the electrical leads. 

7. Pressurize the system to a few h\mdred psi. 

8. Fill the inner dewar and the thermocouple reference dewar 
with nitrogen. 

9. After the bomb has been cooled to the desired temperature 
range evacuate the can containing the bomb and turn on the 
heater to control the temperature. 



91 

10. Take data points about every 0.1°K in the temperature 
region near the transition. 

11. Top off the thermocouple reference dewar and fill the 
cold trap dewar with liquid nitrogen, 

12. Connect the nitrogen dewar for remote transfer. 

13. Close the blow down valve and plug in power cord. 
ik. To prepare for first pressure application open the 

charging solenoid valve to charge the pressure system 
from the 2,000 psi helium storage cylinder. Make s^xce 
that both the intensifier and the separator are cycled 
to the bottom of their stroke. 

15. Close the charging solenoid valve and open the blow- 
down solenoid valve. 

16. Pressurize the system with the separator for the first 
pressure point. 

17. Slowly release the oil pressure in the separator, 

18. Control the temperature with the heater and take 
readings at this pressure. 

19. Activate the pump to the first separator, 

20. Go to the desired pressure and take data. 

21. After a run is complete bleed the pressure down very 
slowly. If the pressure is released rapidly the 
intensifier seals may be seriously damaged. 

The procedure for work between SO'^K and 77°K is 
the same as that outlined above except that the outer dewar 
is filled with liquid nitrogen and a vacuum pump is con- 
nected to the inner dewar. 



I miTWiW I "T^iirn I 1 -|i n Ttla I ■■ 



92 

Procedure for exTPerlment below 60°K. 

For work below 60°K it is necessary to use liquid 
helium and the procedure must be altered. 

The first three steps are the same as listed before. 
^. Fasten the copper thermal shorts to the bomb plug with 
woods metal. 

5. VJoods metal the alternate can, except for bottom 
onto the flange around the bomb. 

6. Connect the copper straps to the bottom of the can and 
seal this onto the can walls. Check for leaks. 

7. Insert the helium transfer line through fitting pro- 
vided in the flange. 

8. Lower the can into the dewar and fasten the top with 
bolts. 

9. Carry the dewars into the pressure room and connect all 
lines and electrical connections. 

10. Pressurize the system to a few hundred psi. 

11. Fill the inner, outer and thermocouple reference dewars. 
Let the system cool to about 77°K, 

12. Transfer the liquid nitrogen out of the inner dewar and 
connect the liquid helium dewar for remote transfer. 

13. Start the transfer. A flow meter and a bubbler can be 
used in the recovery line from the helium cryostat to 
determine the rate at which the transfer is progressing. 

l^J-. Set the micrometer needle valve on the bottom of the 
control panel so that the pressure gauge on the line 



a^inr J usiSmS 



93 

supplying helium for the transfer indicates 3 psi or 
less. The micrometer valve gives fairly fine control, 
and the rate of cool down can be controlled by 
adjusting it. 

15. After the system has cooled appreciably below 77°K 
^ evacuate the can surrounding the bomb. 

16, As the temperature of the bomb reaches the freezing 
temperature of the helium inside it turn on the heater 
on the inlet pressure line and on the top of the bomb. 
Transfer slowly and maintain the top of the bomb 
several degrees above the bottom. Continue this until 
well below the freezing temperature to insure that the 
helium freezes from the bottom to the top of the bomb. 

y-. 17. After the lowest temperature desired is obtained it can 

be maintained or allowed to increase slightly by vary- 
ing the helium transfer rate. 

18. Data can be taken while cooling or warming. 

19. If it is desired to increase the pressure it is neces- 
sary to warm to a temperature greater than the freezing 
temperature of heliiim at the desired pressure. 

20. Pressurize by following 13-1? of the previous section. 

21. Steps 13-18 of this section are then repeated. 



CHAPTER IV 
RESULTS AND CONCLUSIONS 

Introduction 

The data taken in this study fall quite natu- 
rally into two categories. The higher temperature tran- 
sitions (Neel transitions) and the lower temperature 
transitions (Curie transitions). The Neel transitions 
are, as discussed in Chapter II, mainly caused by the 
indirect exchange mechanism and are second order phase 
transitions. The Curie transitions are due to the temp- 
erature variations of the anisotropy energy and are first 
order phase transitions. The first section of this 
chapter presents the data obtained on the Neel transitions 
followed by a comparison with the available theories. The 
second section does the same with the Curie transition 
data. The final section is a discussion of the conclu- 
sions drawn from the results. 



Neel Transitions 



Results 

The Neel transition occurs at about 179°K for 
dysprosivim and was the first one studied. This transition 



94 



IIJU'P I >Mili 1^1— '■■"■■" 



95 

was found to be sharp and reversible. Figure 18 shows a 
run for dysprosium at atmospheric pressure over a temp- 
erature range great enough to show both transitions. In 
determining the Neel transition temperature it was nec- 
essary to traverse only about one degree in temperature. 
The height of the peak was depressed and the location 
shifted by the application of pressure. Figure 19 shows 
a typical run at atmospheric pressure and at 25.000 psi. 
The peak was located in the following way. In the region 
near the maximiim, pairs of temperatures corresponding to 
points of the same inductance on either side of the 
maximum were read from the graph. The locus of the mean 
of these pairs was extrapolated to intersect the experi- 
mental curve. This procedure is indicated in Figure 19* 
It was necessary to rerun the atmospheric pressure peak 
each time the apparatus was reassembled as the location 
of the transition varied somewhat from one assembly to 
the next. This was probably due to small variations in 
the way the thermal gradients arranged themselves each 
time. The shifts were obtained for each pressure at least 
twice and in some cases numerous times. The value of the 
shift was found to be reproducible to within .Oi^°K at a 
given pressure. The pressure shift for dysprosium is 
presented in Figure 20. Each of the determinations fell 
within the datum point shown. The average shift was found 
to be dT^^/dP = -0.44±0.Dl ^K/kilobar. 



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99 
The data for the Neel transition for erbium, 
whioh occurs at about 85°K, were also sharp and reversible. 
The shift for erbium was found to be d Tn/cjP^- O. 26± O.oi 
'^K/kilobar and is presented in Figure 21. 

Comparison of results with theory 

It is possible, as shown in Chapter II, to cal- 
culate the value of dT^^/dP by using the theory of second 
order phase transitions. 

By using the expression, 

dP P ACp 

xfl-here Ao(93 is the height of the thermal expansion anomaly, 
ACp9^ is the height of the specific heat anomaly, and p 95 
is the density, it was possible to calculate dTN/d P = -C?. 45 
°KAilobar for dysprosium. The excellent agreement between 
the calculated value and the experimental value is perhaps 
better than should be expected since the data for Ac< are 
not that accurate. The Ao( is unfortunately unknown for 
erbium so it is not possible to calculate the shift in this 
case. 

The interaction curve of Robinson et al. as shovm 
in Figure 22 would seem to predict a positive value for the 
shift in dysprosium and erbium. As pointed out in Chapter II 
there is some uncertainty as to just what value for the 
transition temperature should be used in this theory. In 
any case our data do not agree with this positive value. 



100 



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102 

McWhan^O has suggested that it would be desirable to con- 
struct an interaction curve for the highest transition 
from which the dependence of the ordering temperature upon 
J, Land S has been eliminated. This can be done by 
dividing T hy ig-\)'^JiJH) , the De Gennes function. ^2 
Such an interaction curve is presented in Figure 23, With- 
out detailed Imowledge of the compressibility of dysprosium 
and erbium in this temperature range it is not possible to 
accurately calculate what the shifts would be according to 
this theory. It can be seen, however, that the proper sign 
is predicted. 3y using Bridgman's room temperature compres- 
sibilities, A=2.74and 2. 63Xio'''cm2Ag for dysprosium and 
erbium respectively, it was possible to calculate dTsz/dP^-d^^ 
*^K/kilobar for dysprosium and dTN^P=-0.:iq OK/kilobar for 
erbium. In view of the uncertainties involved in using 
these compressibility data at lox-x temperatures the agreement 
seems to be quite good. The important point is that not 
only is the proper sign predicted but also the shift for 
erbium is predicted to be smaller than that for dysprosium. 

McVman has calculated ^~~ for all of the experi- 

<a In V ^ 

ments that have previously been done on the rare earths. He 
has found that the points fall on two distinct curves with 
a break between europium and gadolinium. This break is 
probably associated with the fact that the lighter rare 
earths have a modified hexagonal crystal structure. The 
results of McWhans calculations with the data from this work 
on erbium and dysprosium added are presented in Figure 2^. 



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105 
All of the points except gadolinium are calculated for 
Neel temperatures. 



Curie Transitions 

Results 

The ferromagnetic transition for dysprosium was 
much more difficult to work with than the Neel transition. 
The data taken X'fhile cooling were found to differ by about 
lO^K from data taken while warming. Even more disturbing 
was the fact that if the sample was taken to a temperature 
near the transition and held there the inductance values 
would continue to change for a long time. A test was run 
at 77°K and it was found that the inductance was still 
changing slowly at the end of three and a half hours. 
This result made it undesirable to use the method of extra- 
polating the straight line portion of the curve to where it 
intercepted the temperature axis for determining the tran- 
sition temperature. The extrapolation method has been used 
quite successfully to determine pressure shifts of tran- 
sition with some materials. 

It was found that if dysprosium was cooled to 
about 60°Kt far below its transition temperature, and then 
the data were taken while warming the sample the results 
were reproducible and the inflection point of the inductance 
versus temperature curve occurred at 85°K. This value is in 



' 106 

good agreement with other magnetic measurements. -^9 
Several heating rates were tried and were fotmd not to 
change the location of the transition. All of the data 
used in determining dTc/dP xsrere taken at a warming rate 
of about 6 degrees per hour. The experimental curves are 
y) presented in Figure 25 anddTc/^P is given in Figure 26. 

Over the pressure range studied the shift was found to 
bedTc/d P= - 1.2 °K/kilobar. 

The data on erbium for its low temperature 
transition is presented in Figures 27 and 28. As can be 
seen the shape of the curve for cooling data is completely 
different from that of the warming data. The reasons for 
this difference is not understood at the present time. 
A similar situation was found by Adams and Green^'^when 
they studied meteoric iron. The upper peak on the warming 
curve corresponds to an anomaly found at 52*^K by both neu- 
tron diffraction and electrical resistivity experiments. ^9 
From the neutron diffraction work it has been concluded 
that between 80°K and 52°K only the c component of the 
magnetic moment is modulated sinusoidally along the c-axis. 
Sinusoidal modulation of the perpendicular component begins 

' •■ at temperatures lower than 52°K. 

I 

I The loxTOr peak in the warming curve occurs at 20°K 

and would therefore seem to correspond to the ferromagnetic 
transition. The fact that the middle peak on the warming 
curve has not been reported in high field magnetic measure- 
ments is perhaps not too surprising since these experiments 



107 



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111 

do not detect the 52°K transition either. It is curious 
that it has not been seen by the neutron diffraction studies 
since it is about the same magnitude as the other two peaks. 
The upper cooling peak probably corresponds to the 52 degree 
transition with a shift due to rate effects. The tremendous 
peak found at 2^.2°K in the cooling curve is quite sur- 
prising. The pressure shifts of these peaks are presented 
in Figures 29, 30 and 31. The large lower peak in the 
cooling curve showed no shift with pressure while all of the 
others did, A blank run was made in order to insure that 
none of the anomalies that were observed were due to the 
apparatus , 

Comparison of results with theory 

The theory, as developed, by Liu, for the shift in 
the paramagnetic Curie temperature for rare earths can only 
be examined qualitatively here. Unfortunately the data 
were not taken in a way that allows an accurate determina- 
tion of this temperature and even if it had been, the 
compressibilities are not known in this region of temperature. 
Swenson has calculated ^! ^ by using the temperature at 
which the material actually goes ferromagnetic instead of 
the paramagnetic Curie temperature. The results of his 
calculations with the one for dysprosium and erbium from 
this work added are given in Figure 32. The results do not 
show the constant value predicted by Liu's theory but this 
could be due to not having data on the paramagnetic Curie 
temperature. 



112 



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Pressure kilobar 



Figure 29. Pressure shift for middle peak on warming 

data for erbium. 



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for erbium. 



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cooling data for erbium. 



115 






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116 

Conclusions 

The agreement between the experimental and 
thermodynamically calculated shift for the Neel transition 
in dysprosium is quite encouraging. It would be desirable 
to have a more accurate determination of the height of the 
thermal expansion anomaly for dysprosium in order to 
determine if this result is valid. Of course the same 
information is needed for erbium. 

The qualitative predictions based upon the inter- 
action curve, Figure 22, formulated by Robinson et al.^3 
are found to be violated by the data for both dysprosium 
and erbium. 

The interaction curve given by Figure 23 which 
was constructed following a suggestion by McWhan^^ pre- 
dicts shifts of the magnetic transition temperatures that 
are in fair agreement with the experimental ones. This 
curve is based, however, on a phenomenological approach 
and is therefore not as pleasing as one based upon first 
principles. In it no account is taken of the strong 
anisotropic forces which exist in the antiferromagnetic 
and ferromagnetic states of the rare earths. 

There has been, as yet, no explanation given for 
the grouping of the values of -- ^ - J; as given in Figure 2^. 
It is hoped that in the near future some theoretical basis 
can be given for this result. 



117 
While Liu's theory for the pressure dependence 
of the magnetic transitions in the heavy rare earths is 
open to objection on several points, it is based upon 
first principles and is a start in the explanation of 
these phenomena. The present theories indicate that ani- 
^ sotropy has a powerful influence on the magnetic properties 

of the rare earths. This would indicate that some further 
modifications must be made to make Liu's theory more 
realistic. He has based his theory on the premise that the 
indirect exchange mechanism causes the transitions, and 
that the ground state of the system due to the indirect 
exchange is ferromagnetic. This is not in keeping with the 
mainstream of the theoretical development which indicates 
the ferromagnetic transitions are probably caused by the 
anisotropic influences. Of course, as more information 
becomes available on the conduction electron properties 
there will probably be a need to modify the form chosen to 
represent their wave functions. This may quite likely 
bring in more terms with pressure dependence, for, instance 
4-'F(2kFRi) in equation 2.51 may have to be considered. In 
fairness it must be said that the above is speculative 
since no accurately determined data are available to test 
Liu's prediction. Here it is perhaps appropriate to quote 
from a paper by Monfort and Swenson?^ in regard to these 
predictions. "There is need for precise data in order to 
verify this prediction, and except for gadolinium, the 
existing data for the rare earths merely serve to give an 



].. 



118 
order of magnitude for what is a very small effect." 
In addition to precise data ondT/dP It will be necessary 
to have compressibility data in the region of the transi- 
tion. 

If single crystals could be obtained it vxould 
be very useful to do pressure studies on them since 
according to Landau's thermodynamic theory-52 . ^p-^"®"" 
gives the ratio of the magnetic anisotropy energy to the 
magnetic exchange interaction energy. Where ^p is the 
paramagnetic Curie temperature and II refers to measure- 
ments made parallel to the c-axis and _L perpendicular to 
it. This would seem to give a powerful tool to check the 
qualitative predictions of present theories. 

The results for dTN/dP for dysprosium are in 
general agreement with those reported previously and 
shown in Table 1. The dTN/dPfor erbium is new but seems 
to be quite reasonable according to phenomenological 
theories. The results on dTc/dP for dysprosium are in 
agreement with some unpublished data by Swenson but in 
marked disagreement with those of Hobinson et al. ,^3 i^ho 
find a positive shift for low pressures and a negative 
shift at high pressure. The appearance of the new peaks 
in the erbium data at low temperature are not understood 
and it is recommended that further study be given to this 
point. 

In viev; of the fact that anisotropy is important 
and internal strains in the material could set up distur- 
bances in the internal potentials of the material it would 



. 119 
seem very desirable to rerim the low temperature transi- 
tion in erbium with a fully annealed sample to see if 
the previously unreported peaks are still present. 

The accomplishments of this work are listed 
below, 

1. A high-pressure gas system capable of 
generating hydrostatic pressiures at low temperatures 
was designed, constructed and made operational. It 
should be a basic tool in many experiments in future 
years. 

2. A new method for studying pressure shifts 
of magnetic transitions was developed. The necessary 
equipment for doing these studies was constructed and 
brought to the point where they work dependably. 

3. Accurate measurements were made for dT /c/P 
for dysprosium and erbium anddl^/dP for dysprosium. 
Preliminary results were obtained for 6Tc /d F for erbitim. 

^. The way was opened and experiments have 
been suggested which should be very fruitful in further 
checking theories of magnetic transitions. 

5« Publications related to this dissertation: 
A note describing fabrication of a simple, inexpensive 
high pressure gas electrical seal is being prepared; 
also a paper is in progress describing the experimental 
results of this dissertation. 



.--'s. 



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BIOGRAPHICAL SKETCH 

James E. Milton was born May 12, 193^. at 
Floralaj Alabama. In June 1951 » he was graduated from 
Columbia High School in Lake City, Florida. He attended 
the University of Florida intermittently from 1951 - 195^- 
From 1955 until 1958 he served in the United States Army. 
Following his discharge from the Army, he returned to the 
University of Florida. In June 19^0, he received the 
degree of Bachelor of Aeronautical Engineering. From 
September I960, until the present time he has pursued his 
work toward the degree of Doctor of Philosophy. 

James E. Milton is married to the former Mary 
Eleanor Jernigan and is the father of two children. 



This dissertation was prepared under the direction 
of the chairman of the candidate's supervisory committee and 
has been approved by all members of that committee. It was 
submitted to the Dean of the College of Arts and Sciences 
and to the Graduate Council, and was approved as partial 
fulfillment of the requirements for the degree of Doctor of 
Philosophy. 



April, 1966 




u 



f^TLe^. 



Dean, Collerfe//of- Arts and Sciences 



Dean, Graduate School 



Supervisory Committee 



Chairman 






1 I //